Geometric set theory - People · 2019-10-20 · geometric set theory. The purpose of this research direction is to compare transitive models of set theory with respect to their extensional

Geometric set theory

Paul LarsonJindrich Zapletal

vii

Preface

We wish to present to the reader a fresh and exciting new area of mathematics:geometric set theory. The purpose of this research direction is to comparetransitive models of set theory with respect to their extensional agreement anddefinability. It turns out that many fracture lines in descriptive set theory,analysis, and model theory can be efficiently isolated and treated from this pointof view. A particular success is the comparison of various Σ2

1 consequences ofthe Axiom of Choice in unparalleled detail and depth.

The subject matter of the book was rather slow in coming. The initial work,restating Hjorth’s turbulence in geometric terms and isolating the notion of avirtual quotient space of an analytic equivalence relation, existed in rudimen-tary versions since about 2013 in unpublished manuscripts of the second author.The joint effort [63] contained some independence results in choiceless set theorysimilar to those of the present book, but in decidedly suboptimal framework.It was not until the January 2018 discovery of balanced Suslin forcing that theflexibility and power of the geometric method fully asserted itself. The periodafter that discovery was filled with intense wonder–passing from one configura-tion of models of set theory to another and testing how they separate variouswell-known concepts in descriptive set theory, analysis, and model theory. Atthe time of writing, geometric set theory seems to be an area wide open forinnumerable applications.

The authors benefited from discussion with a number of mathematicians, in-cluding, but not limited to, David Chodounsky, James Freitag, Michael Hrusak,Anush Tserunyan, Douglas Ulrich, and Lou van den Dries. The second authorwishes to extend particular thanks to Bernoulli Center at EPFL Lausanne,where a significant part of the results was obtained during the special semesteron descriptive set theory and Polish groups in 2018.

During the work on the book, the first author was supported by grants NSFDMS 1201494 and DMS-1764320. The second author was supported by NSFgrant DMS 1161078.

v

vi PREFACE

Contents

Preface v

1 Introduction 11.1 Outline of the subject . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Equivalence relation results . . . . . . . . . . . . . . . . . . . . . 21.3 Independence: by topic . . . . . . . . . . . . . . . . . . . . . . . 51.4 Independence: by model . . . . . . . . . . . . . . . . . . . . . . . 111.5 Independence: by preservation theorem . . . . . . . . . . . . . . 141.6 Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.7 Notation and terminology . . . . . . . . . . . . . . . . . . . . . . 27

I Equivalence relations 35

2 The virtual realm 372.1 Virtual equivalence classes . . . . . . . . . . . . . . . . . . . . . . 372.2 Virtual structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.3 Classification: general theorems . . . . . . . . . . . . . . . . . . . 422.4 Classification: specific examples . . . . . . . . . . . . . . . . . . . 452.5 Cardinal invariants . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.5a Basic definitions . . . . . . . . . . . . . . . . . . . . . . . 502.5b Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.5c Cardinal arithmetic examples . . . . . . . . . . . . . . . . 542.5d Hypergraph examples . . . . . . . . . . . . . . . . . . . . 57

2.6 Restrictions on partial orders . . . . . . . . . . . . . . . . . . . . 652.7 Absoluteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682.8 Dichotomies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.8a Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 722.8b Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3 Turbulence 813.1 Independent functions . . . . . . . . . . . . . . . . . . . . . . . . 813.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.3 Placid equivalence relations . . . . . . . . . . . . . . . . . . . . . 88

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viii CONTENTS

3.4 Examples and operations . . . . . . . . . . . . . . . . . . . . . . 903.5 Absoluteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.6 A variation for measure . . . . . . . . . . . . . . . . . . . . . . . 97

4 Nested sequences of models 1054.1 Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.2 Coherent sequences of models . . . . . . . . . . . . . . . . . . . . 1064.3 Choice-coherent sequences of models . . . . . . . . . . . . . . . . 109

II Balanced extensions of the Solovay model 117

5 Balanced Suslin forcing 1195.1 Virtual conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 1195.2 Balanced conditions . . . . . . . . . . . . . . . . . . . . . . . . . 1225.3 Weakly balanced Suslin forcing . . . . . . . . . . . . . . . . . . . 128

6 Simplicial complex forcings 1336.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1336.2 Locally countable complexes . . . . . . . . . . . . . . . . . . . . . 1356.3 Complexes of Borel coloring number ℵ1 . . . . . . . . . . . . . . 1376.4 Modular complexes . . . . . . . . . . . . . . . . . . . . . . . . . . 1396.5 Gδ matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1486.6 Quotient variations . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7 Ultrafilter forcings 1577.1 A Ramsey ultrafilter . . . . . . . . . . . . . . . . . . . . . . . . . 1577.2 Fubini powers of the Frechet ideal . . . . . . . . . . . . . . . . . 1587.3 Ramsey sequences of structures . . . . . . . . . . . . . . . . . . . 1607.4 Semigroup ultrafilters . . . . . . . . . . . . . . . . . . . . . . . . 164

8 Other forcings 1698.1 Chromatic numbers . . . . . . . . . . . . . . . . . . . . . . . . . 1698.2 Discontinuous homomorphisms . . . . . . . . . . . . . . . . . . . 1778.3 Automorphisms of P(ω) modulo finite . . . . . . . . . . . . . . . 1798.4 Kurepa families . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1818.5 Set mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1838.6 Saturated models on quotient spaces . . . . . . . . . . . . . . . . 1868.7 Non-DC variations . . . . . . . . . . . . . . . . . . . . . . . . . . 1908.8 Side condition forcings . . . . . . . . . . . . . . . . . . . . . . . . 1918.9 Weakly balanced variations . . . . . . . . . . . . . . . . . . . . . 195

9 Preserving cardinalities 2019.1 The well-ordered divide . . . . . . . . . . . . . . . . . . . . . . . 2019.2 The smooth divide . . . . . . . . . . . . . . . . . . . . . . . . . . 2059.3 The turbulent divide . . . . . . . . . . . . . . . . . . . . . . . . . 2129.4 The orbit divide . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

CONTENTS ix

9.5 The EKσ divide . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2299.6 The pinned divide . . . . . . . . . . . . . . . . . . . . . . . . . . 236

10 Uniformization 23910.1 Tethered Suslin forcing . . . . . . . . . . . . . . . . . . . . . . . . 23910.2 Pinned uniformization . . . . . . . . . . . . . . . . . . . . . . . . 24010.3 Well-orderable uniformization . . . . . . . . . . . . . . . . . . . . 24210.4 Saint Raymond uniformization . . . . . . . . . . . . . . . . . . . 24410.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

11 Locally countable structures 25311.1 Central objects and notions . . . . . . . . . . . . . . . . . . . . . 25311.2 Very Suslin forcings . . . . . . . . . . . . . . . . . . . . . . . . . 26011.3 Iteration theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 26511.4 Locally countable simplicial complexes . . . . . . . . . . . . . . . 270

11.4a Suslin σ-centered complexes . . . . . . . . . . . . . . . . . 27211.4b Suslin σ-linked complexes . . . . . . . . . . . . . . . . . . 278

11.5 Larger graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28011.6 Collapses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28711.7 Compactly balanced posets . . . . . . . . . . . . . . . . . . . . . 292

12 The Silver divide 29912.1 Perfectly balanced forcing . . . . . . . . . . . . . . . . . . . . . . 29912.2 Bernstein balanced forcing . . . . . . . . . . . . . . . . . . . . . . 30512.3 Placid forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31512.4 Existence of generic filters . . . . . . . . . . . . . . . . . . . . . . 320

13 The arity divide 32713.1 m,n-centered and balanced forcings . . . . . . . . . . . . . . . . 32713.2 Preservation theorems . . . . . . . . . . . . . . . . . . . . . . . . 32813.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

14 Other combinatorics 34314.1 Maximal almost disjoint families . . . . . . . . . . . . . . . . . . 34314.2 Unbounded linear suborders . . . . . . . . . . . . . . . . . . . . . 34514.3 Measure and category . . . . . . . . . . . . . . . . . . . . . . . . 34614.4 Definably balanced forcing . . . . . . . . . . . . . . . . . . . . . . 34914.5 F2 structurability . . . . . . . . . . . . . . . . . . . . . . . . . . . 35214.6 The Ramsey ultrafilter extension . . . . . . . . . . . . . . . . . . 355

Bibliography 361

Index 368

x CONTENTS

Chapter 1

Introduction

1.1 Outline of the subject

Geometric set theory is the research direction which studies transitive modelsof set theory with respect to their extensional agreement. It turns out thatnumerous existing concepts in descriptive set theory and analysis have intuitiveand informative geometric restatements, and the geometric point of view makesit possible to readily isolate many new concepts and generalize old ones. Sur-prising parallels appear between areas that to date did not have a discernibleinterface.

The oldest part of the subject deals with Borel equivalence relations on Polishspaces. Here, given an equivalence E on a space X and a configuration {Mi : i ∈I} of transitive models of set theory, one asks whether it is possible for an E-equivalence class to have representatives in some models of the configuration andfail to be represented in others. The answer to this question greatly varies withthe nature of the equivalence relation and the configuration in question, andthe resulting differences can be used with great effect to prove non-reducibilityand ergodicity theorems for various Borel equivalence relations. One notablesuccess in this area is a reformulation of Hjorth’s turbulence in geometric terms,which is much more practicable than the original definition and allows endlessgeneralizations and variations entirely divorced from the original context ofPolish group actions.

Geometric set theory really started to blossom after it was applied in aseemingly entirely different direction: the independence results in choicelessZermelo–Fraenkel (ZF) set theory. This is an area which saw a surge of re-sults in early 1970’s after which the interest in it waned. Both our purposeand methodology are quite different from the early practitioners though. Westudy almost exclusively ZF independence results between Σ2

1 consequences ofthe Axiom of Choice which are intimately connected to various contemporaryconcerns of descriptive set theory and analysis. A very detailed structure ap-pears, with some fracture lines running in parallel to existing combinatorial,

1

2 CHAPTER 1. INTRODUCTION

algebraic, or analytic concepts and other fracture lines showing up in placesquite unexpected. The main geometric motif is the following. Given a Σ2

1 sen-tence φ = ∃A ⊂ X ψ(A), a consequence of the Axiom of Choice in which X is aPolish space and ψ is a formula quantifying only over elements of Polish spaces,and given a configuration {Mi : i ∈ I} of transitive models of set theory withchoice, is there a set A ⊂ X such that in all (or many) models M in the config-uration, A∩M ∈M and A∩M is a witness to φ in the model M? The answerto this question is surprisingly varied and successful in separating consequencesof the Axiom of Choice of this syntactical complexity.

The models of ZF we use for the independence results are nearly exclusivelyof the same form: they are extensions of the symmetric Solovay model by a sim-ply definable σ-closed forcing. As a result, they are all models of DC, the Axiomof Dependent Choices. DC is instrumental for developing the basic concepts ofmathematical analysis, thus highly desirable; at the same time, it is an axiomwhich is difficult to obtain in the symmetric models of 1970’s. If suitable largecardinals are present, essentially identical effects can be obtained in generic ex-tensions of the model L(R) which have a very strong claim to canonicity. Inaddition, given Σ2

1 sentences φ0, φ1, there is usually a quite canonical choicefor a forcing which should generate a model of ZF+DC+¬φ0+φ1. The wholeanalysis of the forcing in question takes place in ZFC, using central concepts ofthe fields related to the Σ2

1 sentences in question.We conclude this brief outline of the area with the observation that the mass

of natural questions that either seem to be treatable using the geometric method,or are generated by the inner workings of the geometric method, is entirelyoverwhelming. This book cannot be more than a relatively brief introductionto the subject. In many directions, our efforts faded due to sheer exhaustionand not because of insurmountable obstacles. We hope to motivate the readerto explore the enormous, inviting expanses of geometric set theory.

1.2 Equivalence relation results

The first group of results deals with the simplest geometric concern. If E is ananalytic equivalence relation on a Polish space, and V [H0], V [H1] are mutuallygeneric extensions of V , is it possible to find an E-class which is represented inboth V [H0] and V [H1] but not in V ? Kanovei [48, Chapter 17], continuing theseminal work of Hjorth, defined an analytic equivalence E to be pinned just incase the answer is negative.

In Chapter 2, we develop the notion of the virtual quotient space for a givenanalytic equivalence relation on a Polish space. A virtual E-class is one which isrepresented in some mutually generic extensions of V ; from a different angle, itis a class which may be represented only in some generic extension, but it doesnot depend on the choice of the generic filter. A formal definition uses the notionof an E-pin and an equivalence on the class of E-pins, which naturally extendsthe analytic equivalence relation E into the transfinite domain–Definitions 2.1.1and 2.1.4. The virtual E-quotient space is then the class of all E-pins up to

1.2. EQUIVALENCE RELATION RESULTS 3

their equivalence. Every Borel structure on the E-quotient space has a naturalvirtual version on the virtual E-quotient space. Virtual structures appear inmany places in this book. The main theorem governing their behavior:

Theorem 1.2.1. (Proposition 2.2.5) Let E be a Borel equivalence relation ona Polish space X. Let M be a Borel structure on the E-quotient space X/E.The natural map from the E-quotient space to the virtual E-quotient space is aΠ1-elementary embedding from M to its virtual version.

To size up the virtual E-quotient space, in Definition 2.5.1 we isolate cardinalinvariants κ(E) and λ(E) of an equivalence relation E: λ(E) is the cardinality ofthe virtual space, while κ(E) is its forcing-theoretic complexity. The cardinalsκ(E), λ(E) respect the Borel reducibility order of equivalence relations, andas such serve as valuable tools for non-reducibility results. Their key feature:for Borel equivalence relations E, these cardinals are well-defined and obey anexplicit upper bound. In particular, the virtual E-quotient space is a set, asopposed to a proper class as occurs for some analytic equivalence relations.

Theorem 1.2.2. (Theorem 2.5.6) Let E be a Borel equivalence relation on aPolish space X. Then κ(E), λ(E) < iω1

.

As a result, the well-known Friedman–Stanley theorem [31] on nonreducibilityof a jump E+ of a Borel equivalence relation E to E is immediately translated tothe Cantor theorem on the cardinality of the powerset: λ(E+) = 2λ(E) > λ(E)–Example 2.5.5. Other similar conceptual and brief nonreducibility results ap-pear as well. The cardinal invariants κ(E) and λ(E) can attain many exoticand informative values, and inequalities between them can be manipulated byforcing. Most of Section 2.5 is devoted to examples of such manipulation. How-ever, the main underlying concept is absolute, which justifies the parlance usedthroughout the book:

Theorem 1.2.3. (Corollary 2.7.3) Let E be a Borel equivalence relation on aPolish space X. The statement “E is pinned” is absolute among all genericextensions.

We stress that the above result holds within the context of ZFC. If one drops theAxiom of Choice but holds on to Dependent Choices, the class of Borel unpinnedequivalence relations will shrink; in the symmetric Solovay model, it shrinks tothe minimal possible extent: only the canonical unpinned relation F2 and theequivalence relations above it in the reducibility order are unpinned, see below.If one drops even the Axiom of Dependent Choices, the class of Borel unpinnedequivalence relations becomes more erratic. We conjecture that it may expandto its largest possible extent: to the class of equivalence relations which are notreducible to Fσ equivalence relations.

As is usual in the realm of equivalence relations, we aim to prove variousdichotomies.

Theorem 1.2.4. (Theorem 2.8.9) Suppose that there is a measurable cardinal.Let E be an analytic equivalence relation on a Polish space X. Exactly one ofthe following occurs:


1. κ(E), λ(E) <∞;

2. Eω1is almost Borel reducible to E.

Theorem 1.2.5. (Corollary 2.8.13) In the symmetric Solovay model: let E bea Borel equivalence relation on a Polish space X. Exactly one of the followingoccurs:

1. E is pinned;

2. F2 is Borel reducible to E.

In Chapter 3, we deal with a more sophisticated geometric concern. Let Ebe an analytic equivalence relation on a Polish space X. Is it possible to findtwo generic extensions V [H0] and V [H1] of V such that V [H0] ∩ V [H1] = Vin which there is an E-class represented in both extensions V [H0] and V [H1],but not in V ? Note that we do not ask the extensions to be mutually generic.In an initial approach, we develop a fruitful method for constructing separatelyCohen-generic extensions V [H0] and V [H1] such that V [H0] ∩ V [H1] = V . Wedefine (Definition 3.1.3) a practical and easily checked notion of independence ofcontinuous open maps between Polish spaces using a notion of a walk reminiscentof Hjorth’s method of turbulence [48, Section 13.1] and prove:

Theorem 1.2.6. (Theorem 3.1.4) Let X be a Polish space and f0 : X → Y0

and f1 : X → Y1 be continuous open maps to Polish spaces. The following areequivalent:

1. the maps f0, f1 are independent;

2. the Cohen forcing PX of nonempty open subsets of X forces V [f0(xgen)]∩V [f1(xgen)] = V .

As one of the corollaries, we obtain a characterization of Hjorth’s turbulence ingeometric terms. This characterization is much more practical than the originalstatement for anyone familiar with the forcing relation.

Theorem 1.2.7. (Theorem 3.2.2) Let Γ be a Polish group acting on a Polishspace X with all orbits meager and dense. The following are equivalent:

1. the action is generically turbulent;

2. V [x]∩V [γ ·x] = V whenever γ ∈ Γ and x ∈ X are mutually generic points.

To support a broad extension of ergodicity results due to Hjorth and Kechris [52,Theorem 12.5], we develop the classes of placid and virtually placid equivalencerelations (the latter including all equivalence relations classifiable by countablestructures) and prove:

1.3. INDEPENDENCE: BY TOPIC 5

Theorem 1.2.8. (Theorem 3.3.5) Let E be the orbit equivalence relation ona Polish space X resulting from a turbulent Polish group action. Let F bean analytic, virtually placid equivalence relation on a Polish space Y and leth : X → Y be a Borel homomorphism of E to F . Then there is an F -classwhose h-preimage is comeager in X.

It turns out that there is a version of turbulence for measure which leads to manyof the same ergodicity results. It uses the notion of concentration of measurequite close to the abstract whirly actions on measure algebras isolated in [33].

Theorem 1.2.9. (Theorem 3.6.2) Let Γ be a Polish group continuously actingon a Polish space X with an invariant Borel probability measure and an invari-ant ultrametric. Suppose that the action has concentration of measure. ThenV [x] ∩ V [γ · x] whenever x ∈ X is a random point and γ ∈ Γ is a generic pointmutually generic with x.

As usual with the forcing reconceptualizations of various notions in descriptiveset theory, we have to show that the resulting notions are suitably absolute andevaluate their complexity–Theorem 3.5.6.

Chapter 4 addresses infinite configurations of transitive models of ZFC. Themost important case is in which there is an infinite inclusion-descending se-quence 〈Mn : n ∈ ω〉 of models. Under suitable, commonly satisfied coherenceassumptions (Definition 4.2.1 or 4.3.1), the intersection

⋂nMn is a model of

ZF or even ZFC. We spend some time developing a flexible technology of con-structing nontrivial coherent sequences as in Theorem 4.3.5. The breakthoughachieved in this chapter is the theorem showing that similar configurations de-tect the distinction between equivalence relations induced as orbit equivalencesof continuous actions of Polish groups, and those which are not reducible toorbit equivalence relations:

Theorem 1.2.10. (Theorem 4.3.6) Let 〈Mn : n ∈ ω〉 be a choice-coherent se-quence of models of ZFC. Let E be an orbit equivalence relation of a continuousPolish group action with a code in the intersection model

⋂nMn. If a virtual

E-class has a representative in every model Mn, then it has a representative inthe intersection model

⋂nMn.

Recall that the canonical equivalence relation not reducible to an orbit equiv-alence relation is E1. It is not difficult to construct a choice-coherent sequenceof models 〈Mn : n ∈ ω〉 and an E1-class which is represented in each model onthe sequence but not in the intersection model

⋂nMn–Example 4.3.10.

1.3 Independence: by topic

Our concerns in the part of the book that deals with independence results inZF+DC can be grouped into several large areas.

Cardinalities of quotient spaces. Mirroring the traditional concerns of de-scriptive set theory, we work on cardinalities of quotient spaces of Borel equiv-


alence relations. Given Polish spaces X,Y and Borel equivalence relations E,Fon each, descriptive set theorists study the question when there can be a Borelfunction h : X → Y which reduces E to F . This line of work has been verysuccessful in the last two decades.

In the ZF+DC context, the E- and F -quotient spaces can have distinctcardinalities, and the existence of a Borel reduction implies the inequality |E| ≤|F |, where we abuse the notation to write |E| for the cardinality of the E-quotient space. On the other hand, the nonexistence of a Borel reduction isoften connected with the possibility that ZF+DC cannot prove the cardinalinequality |E| ≤ |F |. A number of our results deal with the question whethera given Σ2

1 statement is consistent with ZF+DC plus a statement of the type|E| 6≤ |F | for some benchmark Borel equivalence relations E,F identified inSection 1.7.

The most commonly considered quotient space cardinal inequality is |E0| >|2ω| and we spend a great deal of energy proving that various Σ2

1 statements areconsistent with it.

Theorem 1.3.1. The following statements are separately consistent with ZF+DCand |E0| > |2ω|:

1. ([21], Corollary 9.2.5) There is a nonprincipal ultrafilter on ω;

2. (Corollary 9.2.21)there is a discontinuous homomorphism of R to R/Z;

3. (Corollary 9.2.12) given a Borel equivalence relation E on a Polish space,the E-quotient space can be linearly ordered;

4. given a pinned Borel equivalence relation E, |E| ≤ |E0|;

5. (Corollary 11.4.12) given a countable Borel equivalence relation E, E isthe orbit equivalence relation of a (discontinuous) action of Z.

It is an interesting question whether some of the statements above can be com-bined; in general, we have

Question 1.3.2. Are there Σ21-statements φ0, φ1 such that each of them is

consistent with ZF+DC and |E0| > |2ω|, while their conjunction implies inZF+DC that |E0| ≤ |2ω| holds?

We give a similar treatment to certain other well-known quotient cardinal in-equalities. For example,

Theorem 1.3.3. The following statements are separately consistent with ZF+DCand |E1| 6≤ |F | for any orbit equivalence relation F of a continuous Polish groupaction:

1. (Corollary 9.4.8) There is a nonprincipal ultrafilter on ω;

2. (Corollary 9.4.11) there is a discontinuous homomorphism of R to R/Z;


3. (Corollary 9.4.22) given a Borel graph Γ on a Polish space, Γ contains amaximal acyclic subgraph;

4. (Corollary 9.4.10) given a Borel equivalence relation E on a Polish space,the E-quotient space can be linearly ordered;

5. (Corollary 9.4.15) given a pinned orbit equivalence relation E of a contin-uous Polish group action, E has a transversal.

Theorem 1.3.4. The following statements are separately consistent with ZF+DCand |E| 6≤ |F | for any orbit equivalence relation E of a turbulent Polish groupaction and an analytic equivalence relation F classifiable by countable structures:

1. (Corollary 9.3.6) Given a Polish vector space X over a countable field, Xhas a basis;

2. (Corollary 9.3.7) given a Borel graph G on a Polish space, G contains amaximal acyclic subgraph;

3. (Corollary 9.3.10) given a pinned equivalence relation G classifiable bycountable structures, G has a transversal.

There are many interesting open questions left in the area of quotient spacecardinalities. For one, we do not know whether the various independence resultswe obtain can be in general combined in the spirit of Question 1.3.2. There arealso more specific questions. Can the cardinalities of countable Borel equivalencerelations can be manipulated in the context of ZF+DC? For example, are therecountable Borel equivalence relations E,F such that both inequalities |E| < |F |and |F | < |E| are consistent with ZF+DC? In ZF+DC, does the existenceof a nonprincipal ultrafilter on ω introduce new provable inequalities betweenquotient space cardinalities?Locally countable structures. One particularly popular subject of descrip-tive set theory in the last quarter-century has been the study of countable Borelequivalence relations and the various structures on their equivalence classes.Using the methods of this book, one can identify many new fracture lines inthis subject; various selection principles can be meaningfully stratified by theZF+DC provability of implications between them. The following is a samplingof the results we obtain.

Theorem 1.3.5. (Corollary 11.4.17) Let G be an analytic graph on a Polishspace, with uncountable Borel chromatic number. Given a locally finite bipar-tite Borel graph H satisfying the Marks–Unger condition, it is consistent thatZF+DC holds, H has a perfect matching and the chromatic number of G isuncountable.

Theorem 1.3.6. Let G be an analytic hypergraph with all hyperedges finite, ona Polish space X, with uncountable Borel chromatic number. The following areconsistent with ZF+DC plus the statement that the chromatic number of thehypergraph G is uncountable:


1. (Corollary 11.4.12) given a countable Borel equivalence relation E on aPolish space with all classes infinite, E is the orbit equivalence relation ofa (discontinuous) action of Z;

2. (Corollary 11.4.10) given a locally finite acyclic Borel graph Γ on a Polishspace, the chromatic number of Γ is ≤ 3;

3. (Corollary 11.6.7) given a pinned Borel equivalence relation E on a Polishspace, |E| ≤ |E0|.

An important fracture line among Borel locally countable graphs appears at thelevel of the diagonal Hamming graph H<ω which is the product of cliques on2, 3, 4 . . . :

Theorem 1.3.7. The following are consistent with ZF+DC plus the statementthat the chromatic number of H<ω is uncountable:

1. (Corollary 11.7.8) there is a nonprincipal ultrafilter on ω;

2. (Corollary 11.7.10) given a locally finite bipartite Borel graph Γ satisfyingthe Hall’s marriage condition, Γ has a perfect matching;

3. (Corollary 11.7.9) given a Borel equivalence relation E on a Polish spaceX, the E-quotient space is linearly ordered;

4. (Corollary 11.6.3) given a pinned Borel equivalence relation E on a Polishspace, |E| ≤ |2ω|.

Note that in the case of the last consistency result for the case E = E0, thecountable uniformization has to fail, as the uncountable chromatic number ofH<ω implies among other things that E0 has no transversal.Another fracture line occurs at the level of the Hamming graph Hω on ωω whichis the product of infinitely many cliques on ω:

Theorem 1.3.8. (Corollary 11.5.10) It is consistent with ZF+DC that the chro-matic number of the diagonal Hamming graph H<ω is countable while the chro-matic number of the Hamming graph Hω on the Baire space is uncountable.

Model theory. Given a pre-geometry on a Polish structure (for example, thepre-geometry of linear span on a Polish vector space), the axiom of choice yieldsa maximal independent set (a basis for the vector space). This broad strokeerases many fine combinatorial distinctions between the various pre-geometries.In the choiceless context, interesting distinctions become visible.

In our context, it is more convenient to work with Borel matroids, a combi-natorial concept dual to pre-geometries. Matroids from a broad class give riseto balanced forcings (Theorem 6.5.2), providing theorems of the following kind:

Theorem 1.3.9. (Corollary 12.2.13) It is consistent with ZF+DC that thereis a transcendence basis for the reals over Q and there is no diffuse probabilitymeasure on ω.


One traditional fracture line between matroids leads between the modular ones(like the matroid of linearly independent sets) and the non-modular ones (likethe matroid of algeraically independent sets). We develop an abstract notionof modularity in Definition 6.4.1 and, improving on earlier work of [79, 12], weprove:

Theorem 1.3.10. (Corollary 12.3.8) Let X be a Borel vector space over acountable field Φ. Let Y be an uncountable Polish field with a countable subfieldΨ. It is consistent with ZF+DC that there is a Hamel basis for X over Φ butno transcendence basis for Y over Ψ.

Matroid theorists also distinguish between linear matroids (like the matroid oflinearly independent sets) and graphoids (the matroids of finite acyclic subsetsof a given graph). This distinction gives rise to the following theorem:

Theorem 1.3.11. Let G be a Borel graph on a Polish space X. It is consistentwith ZF+DC that G contains a maximal acyclic subgraph and there is no Hamelbasis for the reals.

The proof takes an unexpected turn: we prove that in ZF+DC, existence ofa Hamel basis implies that E1 has a complete countable section, in particular|E1| ≤ |F2|–Corollary 9.4.30. At the same time, existence of a maximal acyclicsubgraph of any Borel graph G is consistent with ZF+DC+ |E1| 6≤ |E| for anyorbit equivalence relation E–Corollary 9.4.22.

In another direction connecting model theory with choiceless ZF+DC argu-ments, we study theories which can have models on the various quotient spaces.In a typical development, given a Borel equivalence relation E on a Polish spaceX and a Fraisse class F of relational structures with strong amalgamation, wecan add a F-structure (a structure all of whose finite induced substructures arein F) on the E-quotient space in a highly controlled way. This results in manytheorems of the following kind:

Theorem 1.3.12. (Corollary 13.3.2) Let E be a Borel equivalence relation ona Polish space X. It is consistent with ZF+DC that there is a tournament onX/E while there is no linear ordering on 2ω/E0.

Combinatorics. A challenging part of set theory deals with chromatic numbersof various natural Borel graphs and hypergraphs. Non-locally countable graphsrequire an entirely different approach. For a sample result, for each number n >0 consider the graph Gn on Rn connecting points of rational Euclidean distance.The work of Schmerl [80] shows that in ZF, the existence of a transcendencebasis for the reals over the rationals implies that these graphs all have countablechromatic number. We can prove the following:

Theorem 1.3.13. It is consistent with ZF+DC that the chromatic number ofG1 is countable while the chromatic number of G2 is uncountable.


In fact, it turns out that the existence of a Hamel basis for R implies in ZF+DCthatG1 has countable chromatic number, while it is consistent with ZF+DC plusthe statement that G2 has uncountable chromatic number (Theorem 12.3.5).The analysis of the graphs Gn becomes more difficult as the number n increases,and the following attractive question is left open for all n > 1:

Question 1.3.14. Let n > 1 be a number. Is it consistent with ZF+DC thatthe chromatic number of Gn is countable while the chromatic number of Gn+1

is uncountable?

Hypergraphs of arity higher than 2 present enormous challenges. To illustratethe possibilities, call a coloring c : R2 → ω an equilateral triangle-free decom-position if there is no equilateral triangle whose vertices are painted the samecolor. Ceder [17] showed that in ZF, the existence of a Hamel basis implies theexistence of an equilateral tringle-free decomposition. The opposite implicationdoes not go through:

Theorem 1.3.15. It is consistent with ZF+DC that the equilateral triangle-freedecomposition exists and yet there is no discontinuous homomorphism betweenPolish groups.

Ultrafilters. The methods of the book can separate various types of ultrafilterson ω and other combinatorial objects in the context of ZF+DC.

Theorem 1.3.16. (Corollary 9.2.5, Proposition 9.2.22 and Corollary ??) InZF+DC, there are no provable implications between the existence of a Hamelbasis and existence of a nonprincipal ultrafilter on natural numbers.

Improving [37], we have

Theorem 1.3.17. (Corollary 14.6.7) In ZF+DC, the existence of a nonprinci-pal ultrafilter on natural numbers does not imply the existence of one which isdisjoint from the summable ideal.

We isolate several ways of adding an ultrafilter to the symmetric Solovay modelin a controlled way, providing ultrafilters with various interesting partition prop-erties: Ramsey ultrafilters and stable ordered union ultrafilters are just twoexamples. However, the comparison of the various models obtained as well asa classification of ultrafilters appearing in each model seems to be beyond ourskill. Many other combinations of implications remain unresolved. For exam-ple, we do not know how to construct a model of ZF+DC in which there is anonprincipal ultrafilter on ω, |E1| ≤ |E0| and yet |E0| 6≤ |2ω|.

Uniformization. Uniformization problems belong to the guiding lights of de-scriptive set theory. One of the most notorious versions of it is countable toone uniformization, the statement that every subset of the plane with nonemptycountable vertical sections contains the graph of a total function. In Chapter 10,

1.4. INDEPENDENCE: BY MODEL 11

greatly improving the methods and results of [56], we develop a satisfactory cri-terion which guarantees that various strong uniformization principles (includingthe countable-to-one uniformization as the humblest special case) hold in nearlyall models under study. Separating the various uniformization principles seemsto be a difficult task beyond the techniques of the present book.

Limitations of the method. The balanced forcing offers a very flexible andpowerful method for obtaining independence results in ZF+DC set theory. How-ever, certain desirable types of independence results are outside of its reach.Certain basic limitations are encapsulated in the following theorems.

Theorem 1.3.18. (Corollary 9.1.2) In every balanced extension of the sym-metric Solovay model, there is no ω1 sequence of distinct Borel sets of boundedBorel rank.

Theorem 1.3.19. (Theorem 14.3.1) In every nontrivial balanced extension ofthe symmetric Solovay model, there is a set of reals without the Baire property.

Theorem 1.3.20. (Theorem 14.1.1) In every balanced extension of the sym-metric Solovay model, there is no maximal almost disjoint family of subsets ofω.

Theorem 1.3.21. (Theorem 14.2.1 simplified) In every balanced extension ofthe symmetric Solovay model, there is no linearly ordered, unbounded set ofelements of ωω in the modulo finite domination ordering.

Thus, the balanced forcing cannot be used to prove for example the consistencywith ZF+DC of the statement that ℵ1 ≤ |E1| but ℵ1 6≤ |E0|. For consistencyresults of this type, one has to reach to the weakly balanced forcing of Sec-tion 5.3. Another apparently difficult question which cannot be resolved usingbalanced forcing is the consistency of ZF+DC plus the statement that there isa non-Lebesgue measurable set of reals, every uncountable set of reals containsa perfect subset, and every set of reals has the Baire property [81]. A similarinteresting question unresolvable by the balanced forcing technique was askedby Marks and Unger: is it consistent with ZF+DC that every set of reals hasthe Baire property and every countable Borel equivalence relation is an orbitequivalence relation of a (discontinuous) action of Z?

1.4 Independence: by model

One appealing way to present the work in this book is to consider several moreor less canonical generic extensions of the Solovay model and list the statementsthat we know hold in them. There are also test problems that we do notknow how to resolve. The reader needs to keep in mind that all the modelsare balanced extensions of the Solovay model W and as such obey the generallimitations described at the end of the previous section. The list below includesonly models that we find of immediate interest for an introduction; there areinnumerable other options.


The Ramsey ultrafilter model. Consider the poset of infinite subsets of ωordered by inclusion and the associated extension of the Solovay model W . Thegeneric filter is identified with a Ramsey ultrafilter, and well-known pre-existingresults [26, 56] show that in a suitable sense, every Ramsey ultrafilter is genericover W for the poset P(ω) modulo finite. The resulting model was investigatedin numerous papers such as [63, 21].

Theorem 1.4.1. In the model W [U ], the following statements hold:

1. ([21], Corollary 9.2.5) |E0| 6≤ |2ω|;

2. (Corollary 9.4.8) |E1| 6≤ |F | for any orbit equivalence relation F ;

3. the E0- and E1-quotient spaces are linearly orderable (Theorem ??), whilethe E2- and F2-quotient spaces (Corollary 14.6.7 and Corollary 14.5.4) donot carry even any tournament;

4. (Example 12.1.9) if E is a Borel equivalence relation and A is a subset ofthe E-quotient space then either |A| ≤ ℵ0 or |2ω| ≤ |A|;

5. (Corollary 14.6.7) every nonprincipal ultrafilter on ω has nonempty inter-section with the summable ideal;

6. (Example 10.5.9) countable-to-one uniformization.

One can consider many variations, adding an ultrafilter which is not Ramsey. Westudy the cases of a stable ordered union ultrafilter and certain other ultrafilters.It is challenging to discern between the resulting models by sentences which donot mention ultrafilters.

There are many open questions about the Ramsey ultrafilter model. Wedo not know how to classify the ultrafilters on ω in it. We do not know ifthere can be Borel equivalence relations E,F such that |E| 6≤ |F | holds in Wand |E| ≤ |F | holds in W [U ]. In particular, we do not know if this can occurfor countable Borel equivalence relations E,F or in the situation where E isthe orbit equivalence relation of a turbulent action of a Polish group and anequivalence F classifiable by countable structures.

The Hamel basis model. Consider the poset P of countable sets of realswhich are linearly independent over the rationals. The generic filter yields aHamel basis B ⊂ R. It does not seem to be easy to provide a simple criterionwhich would guarantee that a given Hamel basis is P -generic over the Solovaymodel. Still, we understand the theory of the model W [B] fairly well:

Theorem 1.4.2. In the model W [B], the following statements hold:

1. (Proposition 9.2.22) E0 has a transversal;

2. (Corollary 9.4.30) E1 has a complete countable section;

1.4. INDEPENDENCE: BY MODEL 13

3. (Corollary 12.3.8) whenever X is an uncountable Polish field and Φ is acountable subfield, there is no transcendence basis for X over Φ;

4. (Corollary 12.2.10) there is no diffuse probability measure on ω;


One can consider a different Polish vector space in place of R and add a basisto it; a natural example is P(ω) viewed with the symmetric difference operationas a vector space over the binary field. The conclusions of the above theoremremain in force. We do not know how to discern between the resulting modelsin general.

If suitable large cardinals exist, one can find a Hamel basis generic overthe model L(R), independently of the size or structure of the continuum–Example 12.4.4. The model L(R)[B] inherits all features quoted in Theo-rem 1.4.2.

Models with a linear ordering. Let E be a Borel equivalence relation on aPolish space X. Consider the poset P of linear orderings of countable subsetsof the E-quotient space, ordered by reverse extension. The generic filter yieldsa linear ordering ≤ on the E-quotient space. The resulting model is a veryhumble extension of the symmetric Solovay model.

Theorem 1.4.3. In the model W [≤], the following statements hold:

1. (Corollary 9.2.12) |E0| > |2ω|;

2. (Corollary 9.4.10) |E1| 6≤ |F | for any orbit equivalence relation F ;

3. (Corollary 13.3.15) there are no discontinuous homomorphism betweenPolish groups;

4. (Corollary 12.2.15) OCA holds;


Other features may depend on the nature of the equivalence relation E. Forexample, if E is pinned, then in the model W [≤] one cannot find a linearordering or even a tournament on the F2-quotient space–Corollary 14.5.5. IfE is classifiable by countable structures, then in the model W [≤] cannot find alinear ordering on quotient spaces of turbulent group action orbit equivalencerelations–Corollary 12.3.9. Many questions remain open. In particular, it is notclear whether in the model W [≤] there is a cardinal inequality between quotientspaces which does not appear already in W . If suitable large cardinals exist andE is pinned, one can find a linear ordering of the E-quotient space genericover the model L(R), independently of the size or structure of the continuum–Example 12.4.6.

One can add other structures to quotient spaces. In particular, one can add atournament instead of a linear ordering. In the resulting model, the E0-quotientspace cannot be linearly ordered–Corollary 13.3.2.


The transversal models. Let E be a pinned Borel equivalence relation ona Polish space X. Consider the poset P of countable subsets of X consistingof pairwise E-unrelated elements, ordered by reverse inclusion. The genericfilter yields an E-transversal T ⊂ X. The study of the resulting model is fairlystraightforward with our methods:

Theorem 1.4.4. In the model W [T ], the following statements hold:

1. (Theorem 12.2.5) the Open Coloring Axiom;

2. (Corollary ??) there is no nonatomic probability measure on ω;

3. (Corollary ??) there is no transcendence basis for R;

4. (Corollary 13.3.10) there is no discontinuous homomorphism between Pol-ish groups;


Other features depend on the equivalence relation E in a rather predictable way.Thus, if E is classifiable by countable structures, then in W [T ], |F | 6≤ |G| holdswhenever F is an orbit equivalence relation of a turbulent group action and G isan equivalence relation classifiable by countable structures. If E is reducible toan orbit equivalence relation, then in W [T ], |E1| 6≤ |G| holds whenever G is anorbit equivalence relation of a turbulent group action. If suitable large cardinalsexist, one can find a transversal generic over the model L(R), independently ofthe size or structure of the continuum. The model L(R)[T ] inherits all featuresquoted in Theorem 1.4.4.

1.5 Independence: by preservation theorem

For a practitioner, it is probably most useful to categorize the results we obtainby the technical preservation theorem for balanced forcing that leads to them.The spectrum of preservation theorems for balanced forcing rivals the properforcing technology. All of the classes considered below (with the exception ofnested balanced and weakly balanced forcings) are closed under countable, fullsupport product.

In the broad class of balanced forcings, a number of preservation theoremsare possible. The balanced extensions of symmetric Solovay model do not addany well-ordered sequences of elements of the Solovay model–Theorem 9.1.1.They do not contain any inequalities of the form |E+| ≤ |E| where E is a Borelequivalence relation and E+ is its Friedman–Stanley jump–Corollary 9.1.5. Cer-tain more sophisticated objects are missing from these extensions, such as max-imal almost disjoint families–Theorem 14.1.1. To overcome these limitations,one has to reach for the weakly balanced forcings. They do not add any sets ofordinals. A prominent example of a weakly balanced forcing which is not bal-anced is one which adds a maximal almost disjoint family with a certain typeof approximations.

1.5. INDEPENDENCE: BY PRESERVATION THEOREM 15

Compactly balanced forcings are those for which the space of balanced classesis naturally organized into a compact Hausdorff space–Definition 9.2.1. Themost prominent example is the poset P(ω) modulo finite–Example9.2.4. Theextensions of the symmetric Solovay model by compactly balanced forcings sat-isfy |E0| > |2ω|–Theorem 9.2.2. It is even true that the chromatic number of acertain Hamming-type graph remains uncountable–Theorem 11.7.2.

Placid forcings are those in which placid virtual conditions exist: theseare virtual conditions p such that for any pair of separately generic extensionsV [H0], V [H1] such that V [H0] ∩ V [H1] = V and for any conditions p0 ∈ V [H0]and p1 ∈ V [H1] such that p0, p1 are both stronger than p, the conditions p0, p1

are compatible–Definition 9.3.1. Thus, the mutual genericity required in the def-inition of balanced forcing is replaced by the weaker demand V [H0]∩V [H1] = V .The most prominent example is the poset adding a Hamel basis to the realswith countable approximations–Example 9.3.5. The extensions of the symmet-ric Solovay model by placid forcings do not contain any inequalities |E| ≤ |F |where E is an orbit equivalence relation of a turbulent Polish group action, andF is an equivalence relation classifiable by countable structures–Theorem 9.3.3.Another attractive property of placid forcings is that they do not add transcen-dence bases for Polish fields–Theorem 12.3.1. There are many other effects (suchas the nonexistence of nonprincipal ultrafilters on ω) which are in fact conse-quences of preservation theorems for the broader class of Bernstein balancedforcings.

Nested balanced forcings are those which contain a sequence of balanced con-ditions for a suitable infinite nested sequence of generic extensions, as in Defini-tion 9.4.1. All compactly balanced forcings are nested balanced–Example 9.4.7.A prominent example of a nested balanced but not compactly balanced forcing isthe poset adding an E0-transversal by countable approximations–Example 9.4.14.The extensions of the symmetric Solovay model by nested balanced forcings donot contain any inequalities |E1| ≤ |E| where E is an orbit equivalence relation–Theorem 9.4.4.

Tethered forcings are those for which, to establish the balance of a pair〈Q, τ〉, it is enough to show that the pair decides virtual conditions of certaincomplexity–Definition 10.1.1. Most balanced forcings are tethered; a prominentexample of an untethered poset is the collapse poset adding an injection of theE0-quotient space to 2ω. In tethered extensions of the symmetric Solovay model,the countable-to-one uniformization (and many other uniformization principles)holds–Theorem 10.3.2. In the extension by the untethered collapse poset, thereis no E0-transversal which means that the countable-to-one uniformization mustfail.

The perfectly balanced forcings are the balanced Suslin forcings for which,whenever {V [Hx] : x ∈ 2ω} is a perfect collection of mutually generic extensionsof V and in each model V [Hx] there is a condition px ∈ P stronger than agiven balanced condition p, there is a condition in P stronger than uncount-ably many conditions px–Definition 12.1.1. Prominent examples include posetsadding ultrafilters, like the poset of infinite subsets of ω ordered by inclusion.The perfectly balanced extensions of the symmetric Solovay model satisfy such


features as the strong Silver dichotomy (Theorem 12.1.6) and a strong versionof OCA (Theorem 12.1.8).

The Bernstein balanced forcings are those posets for which, whenever {V [Hx] : x ∈2ω} is a perfect collection of mutually generic extensions of V and in each modelV [Hx] there is a condition px ∈ P stronger than a given balanced condition p,no condition in P below p is incompatible with more than countably manyconditions px for x ∈ 2ω–Definition 12.2.1. While this definition may look ob-scure at first reading, many posets satisfy it and it has many consequenceswhich are difficult to obtain otherwise. Bernstein balanced forcings add nofinitely additive diffuse probability measures on ω, in particular non nonprinci-pal ultrafilters–Theorem 12.2.3. Among other effects, their generic extensionssatisfy the Open Coloring Axiom–Theorem 12.2.5. For many of them, a fil-ter generic over the model L(R) exists in the theory ZFC plus a suitable largecardinal, independently of the status of the Continuum Hypothesis or similarissues–Theorem 12.4.2. The Bernstein balanced forcings should be viewed asdual to the perfectly balanced forcings, as generic filter for a perfectly balancedforcing and a generic filter for a Bernstein balanced forcing tend to be auto-matically mutually generic–Theorem 12.4.12. Prominent examples of Bernsteinbalanced forcings include all placid forcings and also the forcings adding tran-scendence bases to Polish fields.

3, 2-balanced forcings form a class useful for ruling out discontinuous ho-momorphisms between Polish groups. A poset P is 3, 2-balanced if there are3, 2-balanced virtual conditions p in it; these are virtual conditions such that forany triple V [H0], V [H1], V [H2] of pairwise mutually generic extensions of V andconditions p0 ∈ V [H0], p1 ∈ V [H1], and p2 ∈ V [H2] below p there is a commonlower bound of {p0, p1, p2}–Definition 13.1.1. A prominent example is the forc-ing adding a linear ordering of a given quotient space by countable approxima-tions. 3, 2-balanced extensions of the symmetric Solovay model do not containany discontinuous homomorphisms between Polish groups; in particular, theycontain no nonprincipal ultrafilters on ω–Theorem 13.2.1. Among other effects,they do not contain non-Borel automorphisms between many quotient groups–Theorem 13.2.3. A much stronger form of this property is 3, 2-centeredness, inwhich any collection of three pairwise compatible conditions has a lower bound.Such posets for example do not add linear orderings of quotient spaces exceptfor trivial reasons–Theorem 13.2.7.

There are a number of preservation properties concerned with locally count-able structures. They all consider posets of the form PKL (Definition 11.4.2)and start with a hypothesis on the simplicial complex K. Thus, Theorem 11.4.5shows that if K is Suslin σ-centered then uncountable Borel chromatic numbersof finitary hypergraphs are preserved by the poset PKL and Theorem 11.4.13proves the same for Suslin σ-linked complexes. There are a number of options,and many classes of posets concerned are closed under the countable supportproduct. It seems difficult to get preservation theorems for locally countablestructures without the technology of iterated very Suslin forcings of Chapter 11.

1.6. NAVIGATION 17

1.6 Navigation

The structure of the book is quite complicated and some navigation advice willgreatly enhance the reader’s experience. Part I contains results about reducibil-ity of Borel and analytic equivalence relations and how amalgamation propertiesof models of set theory can be used to disprove it. Part II provides all of thechoiceless independence results. It is difficult to appreciate the methods of PartII without having basic acquiantance with the concepts introduced in Part I.Still, Part I should be viewed as only the gate to Cantor’s paradise, while PartII is already within the gates.

Chapter 2 defines and explores the notion of the virtual quotient space foran analytic equivalence relation E on a Polish space X. On an intuitive level, avirtual equivalence class is one which may only exist in some forcing extensionbut still we already have a sensible calculus for speaking about it.

Section 2.1 provides the basic definitions. If P is a partial order and τis a P -name for an element of X, the name is called E-pinned if its E-classdoes not depend on the generic filter on P . There is a natural equivalence Eon pinned names, and the E equivalence classes are referred to as the virtualE-classes. Section 2.2 shows that if one has a definable structure on the E-quotient space, it is possible to consider the virtual version of that structureon the virtual quotient space. This is a structure Π1-elementarily equivalent tothe original one. Structures such as virtual versions of partially ordered sets ofBorel simplicial complexes will come handy later in the book.

Sections 2.3 and 2.4 deal with the most immediate concern: the classifi-cation problem for the virtual quotient spaces. The virtual equivalence classesshould correspond to some immediately recognizable combinatorial objects, andin many important cases this hope is fulfilled. In a broad class of equivalence re-lations (the pinned equivalence relations of [48, Section 17]) the virtual quotientspace is simply identical to the quotient space. Many jump-type operations onequivalence relations have a natural translation to operations on virtual quo-tient spaces; for example, the Friedman–Stanley jump [31] is translated to apowerset operation–Theorem 2.3.4. Virtual classes of isomorphism relations oncountable structures naturally correspond to uncountable structures of the sametype (Theorem 2.4.5), even though in some cases there may be mysterious vir-tual classes which are not classified in this way [50]. Still, there are many wideopen questions. For example, we do not know how to classify the virtual quo-tient space for the measure equivalence. The virtual space of homomorphism ofsecond countable compact Hausdorff spaces sems to be naturally classified bycompact Hausdorff spaces, but there is no theorem to that effect.

Section 2.5 deals with the next most natural concern: the attempt to sizeup the virtual quotient space in terms of cardinality. The effort is surprisinglysuccessful, generating cardinal invariants of analytic equivalence relations whichcan take all kinds of exotic and informative values. Given an analytic equivalencerelation E, one defines the pinned cardinal κ(E) as the minimal cardinal suchthat all virtual classes of E are represented by names on posets of size < κ. One


also defines λ(E) as the cardinality of the virtual quotient space, and λ(E,P )as the cardinality of the set of virtual classes which are represented by nameson the poset P . These cardinals respect the Borel reducibility order; therefore,they serve as viable tools for the Borel nonreducibility results. In addition, theyinteract well with a number of operations on equivalence relations. In a jarringdevelopment, these cardinals can connect Borel nonreducibility results to hardquestions about singular cardinal arithmetic or similar concerns of transfiniteset theory.

Section 2.6 limits the type of partial orders that can carry a nontrivial E-pinned name. Theorem 2.6.2 shows that, in particular, proper forcings do notcarry any nontrivial E-pinned names. Theorem 2.6.3 shows that as soon asthere are some nontrivial E-pinned names (i.e. E is not pinned), then one canfind them in any partial order collapsing ℵ1. The most curious result is The-orem 2.6.4: if E is Borel reducible to an orbit equivalence relation, then everynontrivial virtual E-class really comes from a collapse of some uncountable car-dinality to ℵ0. Compare this with the efforts of Section 2.4 where some virtualclasses of equivalence relations classifiable by countable structures were clas-sified by uncountable structures to be collapsed down to the countable size.This feature does not persist to arbitrary analytic equialence relations: Ex-ample 2.6.8 produces a nontrivial E-pinned name on the Namba forcing for asuitable equivalence relation E.

Section 2.7 proves two absoluteness results. First, in Corollary 2.7.3 weshow that for a Borel equivalence relation E, the statement “E is pinned”is absolute among all forcing extensions. This greatly simplifies the languageused to formulate later results. A similar statement for analytic equivalencerelations requires large cardinal assumptions. Second, in Theorem 2.7.4, weshow that at least for equivalence relations E reducible to orbit equivalencerelations, transitive inner models of ZFC calculate the pinned cardinal κ(E)predictably: their calculation cannot return a value larger than the actual valueof κ(E) in V .

It is not clear if the class of unpinned Borel equivalence relations allows aneat basis in ZFC. The main result of Section 2.8 shows that the answer isaffirmative at least in the symmetric Solovay model. There, the basis for un-pinned Borel equivalence relations is just {F2} (a former conjecture of Kechris,which nevertheless fails badly in ZFC); for unpinned analytic equivalence rela-tions we isolate the basis {F2,Eω1}–Corollary 2.8.13 and Theorem 2.8.11. Wealso provide a basis for the class of analytic equivalence relations E such thatκ(E) =∞. The basis is {Eω1

}–Theorem 2.8.9.

Chapter 3 restates and greatly generalizes Hjorth’s notion of turbulencein forcing terms. This development shows that nonturbulent equivalence re-lations are in fact parallel to pinned equivalence relations in a very precisesense. The forcing relation encapsulates many distracting estimates needed inthe traditional treatment of turbulence, resulting in a clean and efficient generalcalculus.

Section 3.1 provides the motivating insight–namely, that building pairs of

1.6. NAVIGATION 19

models V [y0], V [y1] where y0 ∈ Y0 and y1 ∈ Y1 are separately Cohen genericelements of their respective Polish spaces such that V [y0]∩V [y1] = V depends ona suitable notion of a walk–Theorem 3.1.4. The comparison of generic extensionsvis-a-vis their intersection is an important concern in the later parts of thisbook. Section 3.2 provides two basic examples. In the first, Hjorth’s notion ofturbulence of group actions is restated in geometric terms–Theorem 3.2.2. Thesecond example provides a complete characterization of those analytic ideals onω such that it is possible to find points x0, x1 ∈ 2ω separately generic over Vsuch that they are equal modulo I and V [x0] ∩ V [x1] = V –Theorem 3.2.4.

Section 3.3 defines two classes of equivalence relations that are simple fromthe turbulent point of view. An equivalence relation E on a Polish space X isplacid if any E-class represented in generic extensions with trivial intersectionis represented in the ground model. E is virtually placid if any virtual E-class represented in such generic extensions is represented as a virtual class inthe ground model. These definitions are preserved under Borel reducibility andunder many natural operations on equivalence relations, as shown in Section 3.4.The main application is the generalization of an ergodicity theorem by Hjorthand Kechris [52, Theorem 12.5] (Theorem 3.3.5): any Borel homomorphismfrom a turbulent equivalence relation to a virtually placid one stabilizes on acomeager set.

Section 3.5 provides a necessary complement to the forcing developmentin this chapter: a theorem asserting that the notions introduced are abso-lute among all generic extensions and evaluating their descriptive complexity–Theorem 3.5.6. Finally, Section 3.6 develops turbulence of group actions formeasure. It turns out that the appropriate concept uses the familiar notion ofconcentration of measure of [73].

Chapter 4 presents one of the entirely novel concepts of geometric set the-ory: that of a coherent nested sequence of models of ZFC, with emphasis on theintersection model. Thus, let 〈Mn : n ∈ ω〉 be an inclusion-decreasing sequenceof transitive models of ZFC and consider the transitive class Mω =

⋂nMn.

A simple coherence condition on the sequence guarantees that Mω is a modelof ZF (Theorem 4.2.9); a coherence condition with a suitable well-ordering pa-rameter guarantees that Vω is a model of ZFC–Theorem 4.3.2. In Section 4.2we develop the technology of building coherent sequences of generic extensions(Theorem 4.2.8) and produce a case in which choice fails in the intersectionmodel. In Section 4.3 we develop a machinery of building choice-coherent se-quences of generic extensions–Theorem 4.3.5. In principle, there is a whole fieldof coherent set theory hiding behind these concepts and the intersection mod-els can serve as new vehicles for obtaining independence results. However, inthis chapter our emphasis is on the connection with Borel equivalence relations.Our main result shows that the choice-coherent sequences of models detect thedifference between E1 and orbit equivalence relations: if E is an orbit equiva-lence relation and C is a virtual E-class represented in each model Vn, then itis represented in Vω–Theorem 4.3.6. This typically fails for E1. The topic isrevisited in Section 9.4 below.


Chapter 5 develops the basic general theory of balanced Suslin forcing.The treatment of Suslin forcing is quite different from that in [7] or similartreatments on the structure of the real line. Our partial orders are typically (butnot necessarily) σ-closed, and serve to add combinatorial objects by explicitcountable approximations in the context of the symmetric Solovay model. Itis important to understand that the study of these partial orders takes placeexclusively within set theory with choice.

Section 5.1 defines the notion of a virtual condition in a Suslin forcing P . Avirtual condition is an element of the completion of the poset P which only existsin some generic extension, but for which one can develop a sensible calculus ofcomparison already in the ground model, quite in parallel to virtual equivalenceclasses developed in Chapter 2. There are natural equivalence and orderingrelations on virtual conditions in P .

Section 5.2 defines the key notion of a balanced pair in a Suslin forcing P .This is a pair 〈Q, τ〉 where Q is a poset and τ is a Q-name for an elementof the forcing P which satisfies a natural amalgamation property in mutuallygeneric extensions. There is a natural notion of equivalence on balanced pairsin P . The motherload feature exploited throughout the rest of the book isthe simple Theorem 5.2.5: every balanced class contains exactly one virtualcondition. The task of understanding a given Suslin forcing P in the contextof geometric set theory is then nearly identical to the classification of balancedvirtual conditions in a given Suslin forcing P . Balanced virtual conditions inbalanced Suslin forcing are quite parallel to master conditions in proper forcing.

Section 5.3 defines weakly balanced Suslin forcing. This is a weakening of thenotion of balanced Suslin forcing designed to produce some effects (such as MADfamilies) which are absent from balanced extensions of the symmetric Solovaymodel. The whole development is quite parallel to balanced Suslin forcing withsuitable quantification adjustments. Some examples of weakly balanced forcingscan be found in Section 8.9. This book’s primary focus is on balanced forcing,and the potential of the weakly balanced variations is left wide open.

Chapter 6 analyzes a class of balanced forcings arising from simplicial com-plexes of various algebraic forms. Namely, let K be a Borel simplicial complexon a Polish space X. Let PK be the poset of countable subsets p ⊂ X suchthat [p]<ℵ0 ⊂ K, ordered by reverse inclusion. Many useful posets in this bookarise in this fashion, and the balance properties of the posets depend on modeltheoretic properties of K. By coincidence or not, most natural examples arematroids. Section ?? introduces the notion of a modular simplicial complex(Definition 6.4.1), parallel to a modular matroid, with a number of examplesand related notions. Another result (Theorem 6.5.2) shows that the poset addinga basis for a matroid is balanced as soon as the matroid satisfies a natural andcommon complexity requirement. Section 6.6 investigates simplicial complexeson quotient spaces, and shows that balanced conditions are naturally foundin the related virtual quotient spaces. Particularly interesting cases concernthe collapse of one quotient cardinality to another, Definition 6.6.2, and thetransversal type posets which to each pair E ⊂ F of Borel equivalence relations

1.6. NAVIGATION 21

on a Polish space X add a maximal set A ⊂ X such that E � A = F � A–Definition 6.6.5.

Chapter 7 analyzes various ways of adding a nonprincipal ultrafilter on ωto a choiceless model. The basic method, the poset of all infinite subsets of ωordered by inclusion, is investigated in Section 7.1. This poset adds a Ramseyultrafilter, and its balanced conditions are classified simply by ultrafilters. Theresulting model has been investigated from many directions previously [21, 90].The next simplest type of ultrafilter is added by a poset with Ramsey sequencesof finite structures, discussed in Section 7.3. It turns out that balanced con-ditions in this type of forcing are classified by ω-sequences of ultrafilters of acertain type. Section 7.2 shows how to add an ultrafilter which is not a P-pointin the simplest fashion: using the quotient poset of P(ω × ω) modulo I whereI is the Fubini power of the Frechet ideal. The interesting development in thiscase is the complexity of balanced conditions–they are classified by ultrafilterson the set of ultrafilters on ω (Theorem 7.2.2). It is interesting to compare thissituation with Section 6.6: there, the balanced conditions were also classified bycomplex sets, but the complexity seemed to be worked into the definitions of theposets from the beginning. Here, the increased complexity appeared withoutinvitation and as a surprise, at least to the authors. Section 7.4 shows how toadd a stable ordered union ultrafilter; the most interesting insight–the balancedconditions are classified by idempotent ultrafilters (Theorem 7.4.7).

Chapter 8 contains the analysis of balanced forcings which are designedto perform a specific task, and from the point of view of the previous chaptersmay seem somewhat ad hoc. In Section 8.1, we consider the problem of addinga coloring of a Borel graph by a fixed number of colors. It turns out that thetask is closely related to the coloring number of graphs introduced by Erdosand Hajnal [25]. If a Borel graph has countable coloring number, then thereis a natural balanced poset (Definition 8.1.1) adding a coloring by countablymany colors (Theorem 8.1.2). The classical examples of such graphs appearon Euclidean spaces as distance graphs (Example 8.1.7 and 8.1.8) and havebeen studied among others by the Hungarian combinatorial school [58]. Addingcolorings to hypergraphs seems to be much trickier business. We show how toadd a countable decomposition of a vector space over a countable field such thatno composant contains three linearly dependent points (Theorem 8.1.12), or acountable decomposition in which each composant is fully linearly independent(Theorem 8.1.19).

Section 8.2 considers the problem of adding a discontinuous homomorphismbetween given Polish groups. This task cannot have a positive resolution ingeneral since many Polish groups have the automatic continuity property, forexample the unitary group [95]. However, we provide a balanced forcing inthe case of abelian groups, one of which is torsion free, and the other divisible(Theorem 8.2.2). Section 8.3 shows that it is possible to force a nontrivial auto-morphism of the Boolean algebra P(ω) modulo finite with a balanced forcing.The forcing consists simply of countable approximations ordered by reverse in-


clusion, and balanced conditions correspond exactly to automorphisms of thealgebra–Theorem 8.3.3. Section 8.4 considers the problem of adding a cofinalKurepa family on a Polish space with a balanced poset. Again, the naturalcountable approximation poset works, and there is a very simple classificationtheorem for the balanced conditions–Theorem 8.4.3. Section 8.5 presents bal-anced forcings for adding set mappings on Polish spaces without nontrivial freesets.

Section 8.6 shows how one can add saturated models of first order theories toE-quotient spaces for a Borel equivalence relation E. An especially interestingcase is adding a saturated limit of a Fraisse class with strong amalgamation;i.e. a linear ordering, a tournament, or similar structures (Theorem 8.6.4). Inthis section, the amalgamation problems familiar from model theory and theamalgamation questions arising in geometric set theory exhibit a particularlystrong affinity.

Section 8.7 shows that there are balanced orderings which produce extensionsof the symmetric Solovay model in which the Axiom of Dependent Choicesfails. The seminal example is the finite-countable poset of Definition 8.7.1 whichintroduces a partition of an uncountable Polish space into two parts, one of themwithout a perfect subset and the other without countably infinite set.

Section 8.8 introduces several balanced posets which in the usual ZFC con-text would be viewed as “side condition” forcings. The general Definition 8.8.1provides a general construction for adding an uncountable set whose intersectionwith every set in I is countable, where I is a given ideal generated by closedsets. Theorem 8.8.6 shows how to add a special type of a Lusin set. The forcingsmostly serve as delimiting examples in Chapter ??.

Section 8.9 provides two examples of weakly balanced forcings, achievingwhat balanced forcing cannot do. In the first case (Theorem 8.9.2) we con-struct a weakly balanced poset which collapses |E| to |E0, for any given Borelequivalence relation E. This breaks the Friedman–Stanley jump barrier whichis one of the basic features of the balanced extensions of the Solovay modelby Corollary 9.1.5. The second example (Theorem 8.9.7) is a weakly balancedposet adding a maximal almost disjoint family, an object one cannot find inbalanced extensions of the Solovay model by Theorem 14.1.1. In general, theweakly balanced arguments are much more difficult than the balanced ones.

Chapter 9 compares cardinalities of quotient spaces in the balanced exten-sions of the symmetric Solovay model W . On the surface, this is an enterprisevery similar to the traditional forcing concerns in the context of the axiomof choice. However, it is important to understand that the non-well-orderedquotient cardinals offer many more opportunities for meaningful independencework and also for surprising ZF results preventing various patterns of cardinalcollapses.

Section 9.1 shows that no new inequalities between well-ordered cardinali-ties and quotient cardinalities are added by balanced forcing. In fact, no newwell-ordered sequences of elements of W are added–Theorem 9.1.1. This re-sult is central for the rest of the book. Section 9.2 provides a very flexible

1.6. NAVIGATION 23

tool for verification of the inequality |E0| > |2ω| in certain classes of balancedextensions. Namely, it turns out that if the balanced classes of the forcing arenaturally organized into a compact Hausdorff space, then the inequality holds inthe P -extension of the symmetric Solovay model–Theorem 9.2.2. The compactHausdorff space in question may certainly be nonmetrizable; in the case of theposet P of infinite subsets of ω ordered by inclusion, it is simply the Cech–Stoneremainder of ω. Among the applications, it is possible to add linear orderingto any Borel quotient space and preserve |E0| > |2ω|–Corollary 9.2.12, or toadd a discontinuous homomorphism from R to R/Z and preserve |E0| > |2ω|as in Corollary 9.2.21. The inequality |E0| > |2ω| can be verified in other wayswhen the balanced classes do not form a compact Hausdorff space, for exam-ple using Theorem 11.4.13 to maintain the chromatic number of the G0-graphuncountable which automatically implies |E0| > |2ω| by Proposition 11.1.6(2).

Section 9.3 shows that the nonreducibility results obtained by Hjorth’s tur-bulence method turn into cardinal inequalities in certain classes of balancedextensions. The main contribution here is Definition 9.3.1, isolating the rel-evant class of placid Suslin forcings. In placid extensions of the symmetricSolovay model, no inequalities of the type |E| ≤ |F | appear where E is the orbitequivalence relation of a turbulent Polish group action, and F is an equivalencerelation classifiable by countable structures. A weakening of placidity is usedin Chapter ?? to rule out combinatorial objects from generic extensions whichhave nothing to do with any group actions, such as nonprincipal ultrafilters.

Section 9.4 provides practical criteria for showing that the classical nonre-ducibility of E1 to any orbit equivalence relation translates into cardinal in-equalities in certain classes of balanced extensions of the Solovay model. Thisturns out to require close study of nested sequences 〈Mn : n ∈ ω〉 of models ofZFC as in Chapter 4. In particular, we must resolve the question whether, fora given Σ2

1 sentence φ which is a consequence of the axiom of choice, one canproduce a set A such that for each n ∈ ω, the set A∩Mn belongs to Mn and isa witness for the sentence φ there. The resulting tricky combinatorial work canbe used to separate for example the existence of a Hamel basis from existenceof maximal acyclic subgraphs in Borel graphs.

Section 9.6 shows that the nonreducibility results between analytic equiva-lence relations obtained by the comparisons of their cardinal invariants κ andλ mostly turn into cardinal inequalities in the Solovay model W , and theseinequalities survive unharmed into the balanced extension.

Chapter 10 deals with the question of whether countable-to-one uniformiza-tion and similar uniformization principles hold in the balanced forcing exten-sions. Section 10.1 provides the a practical criterion on a Suslin forcing P whichimplies that many strong uniformization principles hold in the forcing extensionof the Solovay model by P . It turns out that the criterion is satisfied in a greatnumber of interesting cases, and its verification mostly follows from the carefulclassification of balanced conditions for P . A number of examples appear in Sec-tion 10.5. Sections 10.2, 10.3, and 10.4 study various uniformization principlesin turn and show that they in fact hold in tethered extensions of the symmetric


Solovay model. Lastly, Section 10.5 provides a great number of examples andnon-examples.

Chapter 11 considers the independence results which appear when oneattempts to select one type of structure to each E-class, and not select anothertype; here E is a given countable Borel equivalence relation on a Polish space.The problems considered are motivated by the present practice of descriptivegraph theory. The upshot is a discovery of new and very pertinent fracture linesin the area. The main point of the whole section is that the definable c.c.c.posets have great influence on independence results in ZF+DC, even thoughheretofore they were used only for ZFC independence results.

The fracture lines and their critical objects are defined in Section 11.1. Wedefine two variations of countable Borel chromatic number of hypergraphs onPolish spaces: the Borel σ-finite chromatic number (Definition 11.1.9) and theBorel σ-finite fractional chromatic number (Definition 11.1.18. The relevantclasses of hypergraphs contain minimal objects obtained by the same operationof a skew product (Definition 11.1.1). There are several dichotomies in style verysimilar to the original G0-dichotomy of [54]. A number of examples illustratethe new concepts; among them, there is one obtained from the density versionof the van der Waerden theorem (Example 11.1.20).

Section 11.2 contains information on the basic technical tool needed to proveindependence results on locally finite structures: very Suslin c.c.c. forcing no-tions (Definition 11.2.3) and their iterations. The section is written in such away that specialists on ZFC independence results can use its results for their ownpurposes. The basic result shows that the finite support iterations of very Suslinforcing notions are very Suslin again (Theorem 11.2.7). A long-standing concernof Stevo Todorcevic also makes appearance: for a given Borel simplicial complexK, viewed as a poset with the reverse inclusion ordering, determining whetherthe poset is c.c.c. and calibrating its chain condition carefully. The sectionprovides a novel angle though: the regularity properties of the posets must beexemplified in a simply definable way. Thus, Section ?? discusses posets whichcan be covered by countably many analytic linked pieces and their iterations,and proves an iteration and product preservation theorem. Sections ??, ??,and ?? provide the same treatment for posets which are definably σ-centered,have a definable version of a finitely additive measure, and are σ-liminf-centered,respectively.

Section 11.4 finally produces some independence results in ZF+DC. Thecommon theme: there is a Borel locally countable simplicial complex K on aPolish space, and we study the consequences of the existence of a maximal K-setof a certain type. Adding colorings for various locally countable hypergraphs,for example, can be viewed from this angle. In all cases, the definable forcingproperties of K as a poset ordered by reverse inclusion play central role. Weprove several results resonating with the current descriptive graph theory. LetG be an analytic hypergraph with finite hyperedges, with uncountable Borelchromatic number. Given a countable Borel equivalence relation E, we provethe consistency of ZF+DC plus “E is an orbit equivalence of a Z-action” plus

1.6. NAVIGATION 25

“the chromatic number of G is uncountable”–Corollary 11.4.10. Given a locallyfinite acyclic Borel graph H, we prove the consistency of ZF+DC plus “thechromatic number of H is not greater than three” plus “the chromatic numberof G is uncountable”–Corollary 11.4.12.

Section 11.5 deals with the effects of coloring a larger, perhaps not locallycountable, Borel graphs G on locally countable structures. If the graph G doesnot contain injective homomorphic copies of the bipartite graphs K→ω,ω andKn,ω1 for some n ∈ ω, then the canonical forcing for adding a G-coloring withcountably many colors does not add an E0-transversal (Theorem 11.5.6). If thegraph G contains no injective homomorphic copy of Kn,n, then the canonicalcoloring poset does not add a coloring of the diagonal Hamming graph H<ω withcountably many colors (Theorem 11.5.8). Great examples of such graphs comefrom the Euclidean distance graphs considered at length by Erdos, Hajnal, andKomjath among others. Let D be a Borel set of positive reals, and let G be thegraph on R2 connecting points whose Euclidean distance belongs to the set D.The nature of the set D greatly influences the behavior of the graph. If D is asequence converging to zero, then the graph G falls into the first category above,and so in ZF+DC the chromatic number of G can be equal to ℵ0 without theexistence of an E0-transversal. If the set D is algebraically independent, thenthe graph G falls into the second category and so its chromatic number can beequal to ℵ0 without providing even a coloring to H<ω. On the other hand, ifthe set D contains a dense subgroup of Q, then the existence of a G-coloringwith countably many colors easily implies the existence of an E0-transversal.

Section 11.6 studies the effects of collapses of various Borel cardinals to |2ω|or to E0 on locally finite structures. Perhaps somewhat unexpectedly, collapsinga Borel cardinal such as E0 to 2ω does not yield an E0-transversal; similarly, formany locally countable hypergraphs G, it does not make G have countable chro-matic number. The notion of the fractional chromatic number of a hypergraph(Definition 11.1.18) makes a key apperance in the main result–Theorem 11.6.1.A collapse of a Borel cardinal to E0 has no discernible effect on locally countablestructures, as Theorem 11.6.5 shows.

Section 11.7 revisits the class of compactly balanced forcing. We show thatfor many locally countable hypergraphs G, compactly balanced forcings do notmake G have countable chromatic number. such forcings do not add countablecoloring to a number of locally countable hypergraphs. The notion of Borelσ-finite chromatic number (Definition 11.1.9) is critical here; the diagonal Ham-ming graph H<ω is the most prominent example. In particular, it is consistentwith ZF+DC that there be a nonprincipal ultrafilter on ω, yet the chromaticnumber of H<ω is uncountable.

Chapter 12 presents important two classes of posets which are in a precisesense dual to each other. Both of them offer extensive control over uncountablesubsets of Polish spaces, but they approach the task from exactly orthogonaldirections.

Section 12.1 isolates the classes of perfectly balanced and perfect forcings.The main inhabitant of these classes is the ordering of infinite subsets of natural


numbers ordered by inclusion. However, it turns out that most natural partialorders for adding ultrafilters are perfectly balanced. Extensions of the symmetricSolovay model by perfectly balanced forcings satisfy a strong form of Silverdichotomy: every uncountable subset of a Borel quotient space contains aninjective copy of 2ω (Theorem 12.1.6). Among other effects, they also satisfy astrong form of the Open Coloring Axiom–Theorem 12.1.8.

Section 12.2 presents the dual class of Bernstein balanced forcings. It isvery rich, including the placid posets, but also posets for adding transcendencebases for Polish fields, or posets adding Fraisse structures on Borel quotientspaces. The preservation theorems show in particular that their extensionsdo not contain nonprincipal ultrafilters (Theorem 12.2.3 or satisfy the OpenColoring Axiom (Theorem 12.2.5). The extensions also exhibit properties whichare in certain sense dual to known ZFC independence results. For example, everyunbounded subset of ωω under the modulo finite domination order contains abounded uncountable subset–Theorem 12.2.7.

Section 12.3 contains preservation results that set the class of placid forcingsapart from the much larger class of Bernstein balanced forcings. In particular,placid forcings do not add algebraic transcendence bases to uncountable Polishfields (Theorem 12.3.1). This yields one of the most striking results of this book:in ZF+DC, existence of Hamel bases for Borel vector spaces over countable fieldsdoes not imply existence of transcendence bases for Polish fields over countablesubfields (Corollary 12.3.8). Placid forcings also do not add colorings to a greatnumber of graphs which have countable coloring number in ZFC. Theorem 12.3.5deals with an interesting class of graphs of this type, obtained from algebraiccurves in R2.

The duality between perfectly balanced and Bernstein balanced posets comesto full view in Section 12.4. It turns out that the generic filters for Bernstein bal-anced forcings and perfectly balanced forcings are often automatically mutuallygeneric–Theorem 12.4.12. This is the most precise and elegant way of framingintuitions like “ultrafilters and transcendence bases for fields have nothing todo with each other”. At the same time, filters for Bernstein balanced forcingsgeneric over L(R) exist as a matter of ZFC plus large cardinals, without regardto the structure of continuum issues–Theorem 12.4.2. Thus, we get say genericHamel bases for vector spaces or generic transcendence bases for Polish fields inZFC plus large cardinals. This contrasts with the perfectly balanced ultrafilterposets, where a long line of ZFC independence results may be interpreted asshowing that generic filters for these posets consistently fail to exist.

Chapter ?? presents the classes of m,n-balanced and centered forcings.These classes serve to separate combinatorial objects of different arity of orga-nization. The reader has to keep in mind that the notion of arity used here isquite abstract or even somewhat ineffable.

Section 13.2 contains the main preservation results. The raison d’etre ofthe classes of forcings under discussion is Theorem 13.2.1 which shows thatn + 1, n-balanced forcings do not add discontinuous homomorphisms betweenPolish groups. This means that in particular no nonprincipal ultrafilters on ω

1.7. NOTATION AND TERMINOLOGY 27

are added. A similar effect is produced by Theorem 13.2.3: n + 1, n-balancedforcings do not add non-Borel homomorphisms between various Borel quotientgroups studied in particular by Ilijas Farah [28]. The evaluation of arity comesinto view with Theorem 13.2.4: it turns out that m,n-balanced forcings addno colorings of Polish groups which avoid monochromatic solutions to suitableequations, and the numbers m,n reflect the complexity of the equation. Wealso study the related and much more restrictive class of m,n-centered forcings.A particular success is Theorem 13.2.7 showing that 3, 2-centered forcings donot add linear orderings of Borel quotient spaces except for a trivial reason.Section 13.3 provides a long list of examples and corollaries with particularemphasis on illustrating the differences between the forcing classes introducedin this chapter.

Chapter 14 contains results that for some reason do not fit into the pre-vious chapters; some of our favorite theorems are here. Certain structures areimpossible to add by balanced forcing; MAD families (Theorem 14.1.1) and lin-early ordered unbounded subsets of many partial orderings (Theorem 14.2.1)fall into this category.

In Section 14.3, we prove one of the main constraints on the (nontrivial)balanced extensions of the Solovay model: there must be a set of reals in themwhich does not have the Baire property (Theorem 14.3.1). On the other hand,one does obtain certain models in which every set of real is Lebesgue measurableas per Theorem 14.3.4. An old result of Shelah concerning the consistency ofZF+DC plus every set of reals is Lebesgue measurable plus there is a set of realswithout the Baire property can be presented using this theorem–Example 14.3.5.

In Section 14.4 we treat the very soft class of posets which possess defin-able balanced conditions. While at first sight it may seem that this class con-tains no particularly interesting posets, this is not the case: for example, theposets adding a discontinuous automorphism of P(ω) modulo finite or a count-able complete section to a given pinned Borel equivalence relation belong toit. We show that these posets preserve uncountable chromatic numbers of an-alytic hypergraphs (Theorem 14.4.7) and do not introduce inequalities of theform |E1| ≤ |E| where E is a pinned orbit equivalence relation. Many otherpreservation properties of posets in this class remain unstated.

1.7 Notation and terminology

General. In this book, as in many books on set theory, it is a frequently playedgambit in the proofs to take an elementary submodel of a “large structure”. Thestructure itself is rarely relevant. As a metadefinition, by a “large structure”we always mean the structure 〈H<κ,∈〉 where H<κ is the set of all sets whosetransitive closure has cardinality < κ, and κ is the smallest regular cardinalsuch that H<κ contains all objects named in the proof to that point, and alsotheir powersets. Similarly, the phrase “a large fragment of ZFC” occurs inseveral places in the book. It is never truly informative to analyze precisely


how large a fragment is needed. As a matter of convention, we mean the finitefragment of ZFC including all axioms except the schemas of comprehension andreplacement, and the schemas of comprehension of replacement for all formulasof set theory of 10100 many symbols or fewer.Analytic equivalence relations. A number of concepts and results in thisbook are stated in terms of Borel equivalence relations on Polish spaces. As amatter of basic terminology, if E is an equivalence relation on a set X, a E-transversal is a set T ⊂ X such that T has a singleton intersection which eachequivalence class. If x ∈ X is a point, then [x]E denotes the equivalence classcontaining x. If A ⊂ X is any set, then [A]E denotes the E-saturation of A, theset {x ∈ X : ∃y ∈ A x E y}. The following definition records several benchmarkrelations which are used throughout the book.

Definition 1.7.1.

1. E0 is the Vitali equivalence on 2ω, connecting x, y ∈ 2ω if they differ atonly finite set of entries;

2. E1 is the equivalence relation on (2ω)ω connecting x, y if they differ atonly finite number of entries;

3. E2 is the relation on 2ω connecting x, y if the sum Σ{ 1n+1 : x(n) 6= y(n)}

is finite;

4. F2 is the equivalence relation on (2ω)ω connecting x, y if rng(x) = rng(y);

5. HC is the equivalence relation on P(ω × ω) connecting relations x, y ifeither both are illfounded or fail the axiom of extensionality or fail tohave a maximal element, or they are isomorphic;

6. Eω1is the equivalence relation on P(ω × ω) connecting relations x, y if

either both are illfounded or are not linear orders, or they are isomorphic;

7. if Γ is a coanalytic class of structures on ω invariant under isomorphism,EΓ is the equivalence relation on countable structures on ω connecting twosuch structures if they are both fail to belong to Γ or they are isomorphic;

8. if I is an ideal on ω then =I on 2ω is the equivalence relation connect-ing x, y if {n ∈ ω : x(n) 6= y(n)} ∈ I. There is an identically definedequivalence relation on (2ω)ω.

Borel equivalence relations are naturally ordered by Borel reducibility. In thecase of analytic equivalence relations, the Borel reducibility relation exhibitscertain pathologies, and it is best replaced by some of its strengthenings. Thisis the content of the following definition.

Definition 1.7.2. Let E and F be analytic equivalence relations on respectivePolish spaces X and Y .

1. E is Borel reducible to F , in symbols E ≤ F , if there is a Borel functionh : X → Y such that ∀x0, x1 ∈ X x0 E x1 iff h(x0) F h(x1).


2. E is almost Borel reducible to F , in symbols E ≤a F , if there exists aBorel function h : X → Y and a set Z ⊂ X consisting of countably manyE-classes such that ∀x0, x1 ∈ X \ Z x0 E x1 iff h(x0) F h(x1).

The most permissive comparison of equivalence relations is the one which com-pares just the cardinalities of the quotient spaces. The following abuse of nota-tion is used throughout:

Definition 1.7.3. If E is an equivalence relation on a Polish space X then theE-quotient space is the set of all E-equivalence classes. Moreover, |E| denotesthe cardinality of the E-quotient space.

The Silver dichotomy shows that for a Borel equivalence relation E, |E| is eithercountable or |2ω| ≤ |E|. In the context of the axiom of choice, the latter disjunctimplies that |E| = |2ω| since the quotient space X/E is a surjective image of 2ω.In choiceless context though, the dichotomies satisfying the latter disjunct mayrepresent many different cardinalities, and this is the subject of study of severalsections in this book. It is clear that E ≤ F implies that |E| ≤ |F |, and inthe context of the axiom of dependent choices, E ≤a F and |F | is uncountableimplies that |E| ≤ |F |.Analytic hypergraphs. A hypergraph on a set X is an arbitrary subset Γ ⊂P(X) consisting of nonempty sets. As a matter of convention, our hypergraphsdo not contain any singleton sets. The elements of Γ will be called hyperedgeswhile the elements of X will be called vertices. A hypergraph is finitary if all ofits hyperedges are finite sets. It is a graph if all its hyperedges have cardinalitytwo. A Γ-anticlique is a set A ⊂ X such that Γ∩P(A) = 0. A Γ-coloring is anyfunction with domain X which is not constant on any hyperedge; the elementsof the range of a coloring are referred to as colors. The chromatic number χ(Γ)of the hypergraph Γ is the smallest cardinal number κ such that there is a Γ-coloring with at most κ-many colors. This definition makes sense only in thecontext of the Axiom of Choice in which cardinalities are well-ordered; therefore,χ(Γ) must exist. In a choiceless context, only a limited version is available: wediscern between countable and uncountable chromatic number, and the variousvalues of the countable chromatic numbers.

We will be interested in analytic finitary hypergraphs. Say that Γ ⊂ [X]<ℵ0

is an analytic hypergraph if X is Polish and the set {y ∈ Xω : rng(y) ∈ Γ} isanalytic. The Borel chromatic number of Γ is countable if X can be decomposedinto countably many Borel sets X =

⋃nXn none of which contains all vertices

of a Γ-edge.Forcing. A great part of this book is devoted to forcing. The words forcing,poset, or partially ordered set are treated as synonyms. A condition is anyelement of a partially ordered set. We use the Boolean notation: q ≤ p meansthat the condition q is stronger, more informative than p. Now, let P be apartially ordered set. For a set A ⊂ P and a condition p ∈ P , we write p ≤ ΣAif for every condition q ≤ p there is a condition r ≤ q which is stronger than someelement of A. The formula P φ denotes the statement that every conditionp ∈ P , p φ. Whenever P is a partial ordering, τ is a P -name, and G ⊂ P is


a generic filter, the symbol τ/G denotes the valuation of the name τ accordingto the filter G. Every Polish space X and every analytic set A ⊂ X have acanonical interpretation in every generic extension, which commutes with allusual descriptive set theoretic operations on Polish spaces. For the detailedtheory of interpretations, see [102]; we will use it without explicit mention as iscustomary in the current forcing practice. The interpretations obey two basicabsoluteness rules:

Fact 1.7.4. (Mostowski absoluteness) If M ⊂ N are transitive models ofZF+DC and φ(~p) is a Σ1

1-formula with parameters in M , then M |= φ(~p) ifand only if N |= φ(i~p) where i is the interpretation operation.

Fact 1.7.5. (Shoenfield absoluteness) If M ⊂ N are transitive models of ZF+DCsuch that ω1 ⊂M holds, and φ(~p) is a Π1

2-formula with parameters in M , thenM |= φ(~p) if and only if N |= φ(i~p) where i is the interpretation operation.

In particular, intepretations of analytic equivalence relations are equivalencerelations again, and interpretations of Polish groups and continuous Polish groupactions on Polish spaces are Polish groups and continuous Polish group actionsagain. Mutual relationships between forcing extensions of ZFC are captured inthe following facts:

Fact 1.7.6. If V is a model of ZFC, V [G] is a forcing extension of V , and Mis a model of ZFC such that V ⊂ M ⊂ V [G], then M is a forcing extension ofV and V [G] is a forcing extension of M .

The book is loaded with product forcing notions. This section provides basicinformation on products.

Fact 1.7.7. [45, Lemma 15.9] Let P,Q be posets and in some generic extension,let G ⊂ P,H ⊂ Q be filters separately generic over the ground model. Thefollowing are equivalent:

1. G×H ⊂ P ×Q is a filter generic over the ground model;

2. G ⊂ P is generic over the model V [H].

In the affirmative case, V [G] ∩ V [H] = V .

If G × H ⊂ P × Q is a filter generic over the ground model, we say that thefilters G,H (or their generic extensions) are mutually generic.

The following humble observation greatly simplifies the methodology of prod-uct forcing. It says that mutual genericity of forcing extensions can be char-acterized without an appeal to the specific generic filters and posets that wereused to obtain the extensions, and indeed without any appeal to forcing at all.

Proposition 1.7.8. Let n ∈ ω be a number and {Pi : i ∈ n} be posets. Let{Gi : i ∈ n} be filters separately generic over the ground model V over therespective posets. The following are equivalent:


1.∏iGi ⊂

∏i Pi is a filter generic over V ;

2. whenever {ai : i ∈ n} are subsets of the ground model in the respectivemodels V [Gi] such that

⋂i ai = 0 then there are sets {bi : i ∈ n} in the

ground model V such that for all i ∈ n ai ⊂ bi and⋂i bi = 0 holds.

Proof. Suppose first that (1) holds. Move to the ground model V . Suppose that{ai : i ∈ n} are Pi-names for subsets of the ground model and 〈pi : i ∈ n〉 ∈

∏i Pi

is a condition forcing that⋂i ai = 0. Let bi = {x ∈ V : ∃p′ ≤ pi p

′ Pi x ∈ ai}for each i ∈ n. It is immediate that

⋂i bi = 0 holds, and for all i ∈ n and

pi Pi ai ⊂ bi holds. (2) then follows by the forcing theorem.Suppose now that (2) holds. To confirm (1), suppose towards a contradic-

tion that it fails. There must be an open dense set D ⊂∏i Pi in the ground

model such that∏iGi ∩D = 0. For every number i ∈ n, in the model V [Gi]

consider the set ai = {〈pj : j ∈ n〉 ∈ D : pi ∈ Gi} ⊂∏j Pj . The contradictory

assumption gives that⋂i ai = 0; the assumption (2) yields sets bi ⊂

∏i Pi in

the ground model such that ∀i ∈ n ai ⊂ bi and⋂i bi = 0. In each model V [Gj ],

a genericity argument shows that there must be a condition rj ∈ Gj such thatevery tuple 〈pi : i ∈ n〉 ∈ D with pj ≤ rj must in fact belong to the set bj . Usethe density of the set D to find a tuple 〈pi : i ∈ n〉 ∈ D such that for all i ∈ n,pi ≤ ri holds. Then 〈pi : i ∈ n〉 ∈

⋃i bi, contradicting the choice of the sets bi

for i ∈ n.

The following corollary is easy to prove without the proposition, but its presentproof is much more appealing:

Corollary 1.7.9. Let n ∈ ω be a natural number. Suppose that {Pi : i ∈ n} areposets, {Qi : i ∈ n} are posets,

∏iGi ⊂

∏i Pi is a generic filter, and for each

i ∈ n Hi ∈ V [Gi] is a filter generic over V on the poset Qi. Then∏iHi ⊂

∏iQi

is a filter generic over V .

Proof. The criterion (2) of Proposition 1.7.8 is preserved when passing to smallermodels, in particular from V [Gi] to V [Hi].

The final remark on the product forcing provides a perfect set of genericsover any countable model–a standard trick which comes handy in several placesin the book.

Proposition 1.7.10. Let M be a countable transitive model of set theory andP ∈ M be a poset. Then there is a continuous map h : 2ω → P(P ) such thatfor every finite set a ⊂ 2ω, the product

∏x∈a h(x) is a filter on the poset P |a|

generic over the model M .

Proof. Let 〈Dn : n ∈ ω〉 be an enumeration of all open dense subsets of variousfinite powers of P which appear in the model M . By induction on m ∈ ω buildmaps gm : 2m → P satisfying the following demands:

• g0 : {0} → P is arbitrary;


• for every m ∈ ω, every s ∈ 2m and every t ∈ 2m+1 such that s ⊂ t,gm+1(t) ≤ gm(s) holds;

• for every m ∈ ω and every n ≤ m, if Dn ⊂ P k is a set for some k ≤2m+1 and 〈ti : i ∈ k〉 is a sequence of distinct elements of 2m+1, then〈gm+1(ti) : i ∈ k〉 ∈ D holds.

This is easy to arrange as the last item requires meeting only finitely many opendense sets. In the end, let h : 2ω → P(ω) be the continuous function definedby setting h(x) to be the filter generated by the set {gm(x � m) : m ∈ ω}, andcheck that the demands of the theorem are satisfied.

There are several standard forcing notions used throughout:

Definition 1.7.11.

1. If X is a set then Coll(ω,X) is the poset of finite partial functions fromω to X ordered by reverse extension;

2. if κ is a cardinal then Coll(ω,< κ) is the finite support product of theposets Coll(ω, α) for all α ∈ κ;

3. if X is a topological space then PX is the poset of all nonempty opensubsets of X ordered by inclusion. If X is in addition Polish, then xgendenotes the PX -name for the generic point of PX–the unique element ofthe interpretation of the Polish space X in the PX -extension which belongsto all open sets in the generic filter.

The collapse poset obeys an important factoring rule:

Fact 1.7.12. [45, Corollary 26.11] Let λ be a cardinal and let P be a poset ofsize < λ. Suppose that G ⊂ Coll(ω, λ) is a generic filter and in V [G], H ⊂ P isa filter generic over V . Then there is a filter K ⊂ Coll(ω, λ) generic over V [H]such that V [G] = V [H][K].

Let κ be an inaccessible cardinal. The symmetric Solovay model derivedfrom κ is obtained as follows. Let G ⊂ Coll(ω,< κ) be a generic filter and in

V [G], form the model HODV [G]V ∪P(ω) of all sets hereditarily definable from reals

and elements of the ground model. This is the symmetric Solovay model; wewill always denote it by W , neglecting the dependence on κ and the filter G inthe notation. The theory of the Solovay model has been thoroughly investigatedthroughout the years. We note the following:

Fact 1.7.13. [87] In W , ZF+DC holds, every set has the Baire property andis Lebesgue measurable, and there is no uncountable well-ordered sequence ofdistinct Borel sets of bounded Borel rank.

During the investigation of the symmetric Solovay model, the following technicalfact and its corollary about the symmetric Solovay model will be used withoutmention.


Fact 1.7.14. [45, Section 26] Let κ be an inaccessible cardinal and let W be thesymmetric Solovay extension of V associated with κ. Then

1. every set in W is definable from parameters in V ∪ 2ω;

2. in W , whenever M is a generic extension of V by a partial order of size< κ then W is a symmetric Solovay extension of M .

Corollary 1.7.15. Suppose that κ is an inaccessible cardinal, X is a Polishspace, φ(x, ~y) is a formula of set theory with all free variables displayed, and ~pis a sequence of sets of the same length as ~y. The following are equivalent:

1. Coll(ω,< κ) ∃x ∈ X φ(x, ~p);

2. there exist a poset R of size < κ and an R-name σ for an element of Xsuch that R Coll(ω,< κ) φ(σ, ~p).

Many models of set theory we investigate are extensions of the symmetric Solo-vay model by a suitably definable, typically Suslin, forcing. Since the instru-mental properties of the Suslin forcing in question may not be absolute betweenforcing extensions, we use the following key convention to shorten the statementsof the results we obtain.

Convention 1.7.16. Let ψ be a property of partial orders, and let φ be asentence in the language of set theory.

1. For a Suslin partial order P and an inaccessible cardinal κ, the statement“P is ψ below κ” means that Vκ |= ψ(P ) holds in every forcing extension”.The statement “P is ψ cofinally below κ” means that Vκ |=every forcingextension has a further extension in which ψ(P ) holds”.

2. The phrase “In ψ extensions of the Solovay model, φ holds” denotes thefollowing long statement. Let P be a Suslin forcing and let κ be aninaccessible cardinal such that P is φ below κ. Let W be a symmetricSolovay model derived from κ. Let G ⊂ P be a filter generic over W .Then W [G] |= φ holds.

3. The phrase “In cofinally ψ extensions of the Solovay model, φ holds”denotes the following statement. Let P be a Suslin forcing. Let κ be aninaccessible cardinal. Suppose that P is ψ cofinally below κ. Let W be asymmetric Solovay model derived from κ. Let G ⊂ P be a filter genericover W . Then W [G] |= φ holds.

In several cases, we will need the basics of the stationary tower forcing.

Definition 1.7.17. [64] Let κ be an inaccessible cardinal. The symbol Qκdenotes the (countably based) stationary tower up to κ. That is, Qκ consists ofsets S such that S ⊂ [dom(S)]ℵ0 is stationary and dom(S) ∈ Vκ. The orderingis defined by T ≤ S if dom(S) ⊂ dom(T ) and {x ∩ dom(S) : x ∈ T} ⊂ S. IfG ⊂ Qκ is a generic filter, j : V → M denotes the generic ultrapower derivedfrom G.


Fact 1.7.18. [64] Let κ be an inaccessible cardinal, G ⊂ Qκ be a generic filter,and j : V →M be the generic embedding.

1. If κ is a Woodin cardinal in the ground model, then κ = ωV [G]1 and Mω ⊂

M in V [G]. In particular, M is well-founded.

2. If κ is a weakly compact Woodin cardinal in the ground model, then forevery z ∈ 2ω in V [G] there is a Woodin cardinal λ < κ such that G∩Qλ isgeneric over V and z ∈ V [G∩Qλ]. In particular, the model W = V (RV [G])is a symmetric Solovay extension of V derived from κ.

As a matter of notation, if the generic ultrapower model M is well-founded, itis always identified with its transitive collapse.

Part I

Equivalence relations

35

Chapter 2

The virtual realm

There are many quotient structures in mathematics. It turns out that a typicalquotient structure allows a useful canonical extension to its virtual version.The purpose of this chapter is to lay the foundations of the theory of virtualstructures.

2.1 Virtual equivalence classes

The basis of any quotient structure of interest in the present book is a Polishspace X. A quotient structure worth its salt will also use an equivalence relationon the underlying Polish space E. In this section we indicate how to extend theE-quotient space in a canonical way to a potentially much larger set or class.

Definition 2.1.1. [48, Definition 17.1.2] Let E be an analytic equivalence re-lation on a Polish space X. Let P be a poset and τ a P -name for an element ofX.

1. The name τ is E-pinned if P × P τleft E τright;

2. if τ is E-pinned, then the pair 〈P, τ〉 will be called an E-pin.

The definition may puzzle a novice reader. Its meaning is best illustrated bythe following proposition. An E-pinned name is one which in all forcing exten-sions points at the same E-class, even though that E-class may not have anyrepresentative in the ground model.

Proposition 2.1.2. Let E be an analytic equivalence relation on a Polish spaceX. Let P be a poset and τ a P -name for an element of X. The following areequivalent:

1. 〈P, τ〉 is an E-pin;

2. in every forcing extension, if G0, G1 ⊂ P are filters separately generic overthe ground model, then τ/G0 E τ/G1 holds.

37

38 CHAPTER 2. THE VIRTUAL REALM

Proof. (2) immediately implies (1) by considering the P × P extension. Tosee how (1) implies (2), suppose that V [H] is a forcing extension and in V [H]there are filters G0, G1 ⊂ P separately generic over V . Let G2 ⊂ P be a filtergeneric over V [H]. By the product forcing theorem, G0, G2 ⊂ P are mutuallygeneric filters, and so are G1, G2 ⊂ P . Applying the assumption (1), we see thatτ/G0 E τ/G2 E τ/G1, so τ/G0 E τ/G1 by the transitivity of the equivalencerelation E.

The following is the archetypal example of a non-trivial pinned name.

Example 2.1.3. Consider the poset P = Coll(ω, 2ω) and its name τ for thegeneric surjection from ω to (2ω)V . The name τ is F2-pinned since no matterwhich generic filter G ⊂ P one selects, the range of τ/G is the same: it is the set(2ω)V . Clearly, it is a nontrivial name since the set 2ω is uncountable, so thereis no ground model element of (2ω)ω that can enumerate it and be equivalentto τ .

Given an analytic equivalence relation E on a Polish space X, the E-pinnednames form a seemingly inexhaustible and complicated class. However, this classadmits a natural equivalence relation which usually greatly clarifies matters:

Definition 2.1.4. Let E be an analytic equivalence relation on a Polish spaceX. Suppose that 〈P, τ〉 and 〈Q, σ〉 are E-pins. Define 〈P, τ〉 E 〈R, σ〉 if P ×Q τ E σ.

Proposition 2.1.5. The relation E is an equivalence. If P,Q are posets withE-pinned names τ, σ on them, the following are equivalent:

1. 〈P, τ〉 E 〈Q, σ〉;

2. in every forcing extension, if G ⊂ P and H ⊂ Q are filters separatelygeneric over the ground model, then τ/G E σ/H holds.

Proof. E is clearly symmetric and reflexive by its definition. To see the tran-sitivity, suppose that 〈P0, τ0〉 E 〈P1, τ1〉 E 〈P2, τ2〉. This means that P0 ×P1 × P2 τ0 E τ1 E τ2, and by the transitivity of the equivalence relation E,P0×P1×P2 τ0 E τ2. By the Mostowski absoluteness between the P0×P1×P2

extension and P0 × P2 extension, it is the case that P0 × P2 τ0 E τ2 and con-sequently 〈P0, τ0 E 〈P2, τ2〉.

Now, (2) immediately implies (1) by considering the P × Q extension. Tosee how (1) implies (2), suppose that V [K] is a forcing extension and in V [K]there are filters G ⊂ P,H ⊂ Q separately generic over V . Let H ′ ⊂ Q be afilter generic over V [K]. By the product forcing theorem, G,H ′ are mutuallygeneric filters, and so are H,H ′. Applying the assumption (1), we see thatτ/G E σ/H ′ E σ/H, so τ/G E σ/H by the transitivity of the equivalencerelation E.

Definition 2.1.6. Let E be an analytic equivalence relation on a Polish spaceX.

2.2. VIRTUAL STRUCTURES 39

1. The E-classes are referred to as the virtual E-classes;

2. if z is a virtual E-class and in some generic extension V [G] y is an E-class, we say that y is a realization of z if for some (equivalently, all)representatives 〈P, τ〉 ∈ z and x ∈ y, V [G] |= P τ E x holds.

The following proposition is used throughout this book. It says that E-classesrepresented in mutually generic extensions must be realizations of a virtual E-class in the ground model.

Proposition 2.1.7. Let E be an analytic equivalence relation on a Polish spaceX. Let P0, P1 be partial orders and G0 ⊂ P0 and G1 ⊂ P1 be mutually genericfilters. If x0 ∈ V [G0] and x1 ∈ V [G1] are E-equivalent points then [x0]E is therealization of some virtual E-class from the ground model.

Proof. Suppose that p0 ∈ P0, p1 ∈ P1, τ0 is a P0-name and τ1 is a P1-name suchthat 〈p0, p1〉 τ0 E τ1. It immediately follows that τ0 must be an E-pinnedname on the poset P � p0 so p0 forces [τ0]E to be the realization of the virtualE-class represented by the pair 〈P � p0, τ0〉.

The main question surrounding the virtual E-classes is whether they can beclassified in some informative way. Is there a proper class of virtual E-classesor just a set? If there is just a set, what is its cardinality? Do virtual E-classescorrespond to some more tangible combinatorial objects? This chapter containsmany good answers to similar questions, even though many problems remainunsolved.

2.2 Virtual structures

It is now possible to define virtual versions of quotient structures on Polishspaces.

Definition 2.2.1. An analytic quotient structure is a tupleM = 〈X,E,Ri : i ∈ω, fj : j ∈ ω〉 where

1. X is a Polish space;

2. E is an analytic equivalence relation on X;

3. for every i ∈ ω, Ri ⊂ Xni is an analytic relation which is invariant underE;

4. for every j ∈ ω, fj ⊂ Xmj+1 is an analytic relation which is invariantunder E, and in the E-quotient space it is a graph of a function.

The quotient structure M is Borel if all the relations above including E areBorel.


There are many popular examples of analytic quotient structures. If 〈G, ·〉is a Polish group and H ⊂ G is an analytic normal subgroup, one can form thequotient group G/H. If 〈X,≤〉 is an analytic partial ordering, one can form theseparative quotient under the assumption that the quotient equivalence relationis analytic. Embeddability of countable structures forms an ordering on thequotients space of all countable structures modulo the equivalence relation ofbiembeddability etc.

If M = 〈X,E,Ri : i ∈ ω, fj : j ∈ ω〉 is an analytic quotient structure, thenwe write M∗ = 〈X∗, R∗i : i ∈ ω, f∗j : j ∈ ω〉 for its associated structure on theactual quotient space X/E of all E-classes.

Definition 2.2.2. Let M = 〈X,E,Ri : i ∈ ω, fj : j ∈ ω〉 be an analytic quo-tient structure. The virtual version of M is the tuple M∗∗ = 〈X∗∗, R∗∗i : i ∈ω, f∗∗j : j ∈ ω〉 where

1. X∗∗ is the set or class of all virtual E-classes;

2. for each i ∈ ω, R∗∗i is the relation on X∗∗ of arity ni given by 〈〈Qk, τk〉 : k ∈ni〉 ∈ R∗∗i if

∏kQk 〈τk : k ∈ ni〉 ∈ Ri;

3. for each j ∈ ω, f∗∗j is the relation on X∗∗ of arity mj + 1 given by

〈〈Qk, τk〉 : k ∈ mj + 1〉 ∈ f∗∗j if∏kQk 〈τk : k ∈ mj + 1〉 ∈ fj .

The first proposition says that Definition 2.2.2 is sound and that we receivea structure with the same signature as the original one:

Proposition 2.2.3. The definition of R∗∗i and f∗∗j does not depend on thechoice of representatives of the virtual classes. Moreover, fj is a graph of a(total) function.

Proof. The statements “Ri, fj are E-invariant relations” and “fj is a graph ofa total function in the quotient” are Π1

2; therefore, they are absolute betweenV and all forcing extensions. The first sentence of the proposition immediatelyfollows. For the second sentence, suppose that the function fj has arity mj and〈〈Qk, τk〉 : k ∈ mj〉 is a tuple of E-pins. Let Q =

∏kQk and let τ be a Q-name

for an element of X such that fj(τk : k ∈ mj , τ) is forced to hold. Since in theQ×Q-extension, fj is a graph of a function on the quotient and the names τkfor k ∈ mj are E-pinned, it follows that the name τ is E-pinned as well. Clearly,the tuple 〈〈Qk, τk〉 : k ∈ mj , 〈Q, τ〉〉 belongs to f∗∗j and the second sentence ofthe proposition follows.

In the case of an analytic quotient structure, it is possible that its virtualversion is a proper class. In particular, it is possible to express the wholeordinal axis as an isomorph of a virtual version of an analytic quotient structure–Example 2.4.6. However, if the equivalence relation E is Borel then the virtualversion is a set of size < iω1

by Theorem 2.5.6. In any case, it is possible tostratify M∗∗ into set sized pieces:

2.2. VIRTUAL STRUCTURES 41

Definition 2.2.4. LetM = 〈X,E,Ri : i ∈ ω, fj : j ∈ ω〉 be an analytic quotientstructure. M∗∗κ is the substructure of M∗∗ consisting of the virtual E-classesrepresented by names on Coll(ω, κ).

A proof identical to that of Proposition 2.2.3 shows thatM∗∗κ is closed underall functions ofM∗∗ so it is truly a substructure ofM∗∗. An elementary name-counting argument shows that for each infinite κ, the structure M∗∗κ has sizeat most 2κ. If κ ≤ λ are cardinals then Coll(ω, κ) is regularly embedded inColl(ω, λ) so M∗∗κ ⊆M∗∗λ . Every poset is regularly embedded in Coll(ω, κ) forsome κ so M∗∗ decomposes into a monotone union

⋃κM∗∗κ .

For each analytic quotient structure M, there is a canonical embeddingπ : M∗ →M∗∗ which maps each class [x]E to the virtual E-class of 〈Q, x〉 fora trivial poset Q. The most important fact about the virtual structures is thatthere is some degree of elementarity:

Proposition 2.2.5. The canonical embedding π : M∗ →M∗∗ is Π1-elementary.

Proof. Let κ be a cardinal, and let G ⊂ Coll(ω, κ) be a generic filter. In themodel V [G], let χ : (M∗∗κ )V → (M∗)V [G] be the map sending a virtual E-class in

(M∗∗κ )V to its realization. Thus, we have maps (M∗)V π→ (M∗∗κ )Vχ→ (M∗)V [G].

The composition χ ◦ π sends each equivalence class in V to its interpretation inV [G].

Let φ be a Π1 formula in Lω1ω logic with possible parameters such thatM∗ |= φ. The statement M∗ |= φ is a Π1

2 sentence about the structure M,and by Shoenfield absoluteness it transfers from the ground model V to thegeneric extension V [G]. Thus, the map χ ◦ π is a Π1 elementary embeddingfrom (M∗)V to (M∗)V [G]. It follows that the map π : M∗ → M∗∗κ in V mustbe a Π1-elementary embedding. Since M∗∗ is an increasing union

⋃κM∗∗κ , the

proposition follows.

In particular, if the original quotient structure was a group, a partial order,or an acyclic graph, its virtual version maintains these properties. However,it is important to understand that the embedding does not have to be Σ2-elementary, so virtual versions of connected graphs may become disconnected,virtual versions of divisible groups may not be divisible anymore, and virtualversions of nonatomic partial orders may have atoms.

Example 2.2.6. Let R be the relation on X = (2ω)ω defined by x0 R x1 ifrng(x0) ⊆ rng(x1). Clearly, the tuple 〈X,F2, R〉 is an analytic quotient struc-ture. The relation R∗ is a partial order on X∗ without largest element. Atthe same time, the relation R∗∗ on X∗∗ does have the largest element, namelythe F2-pin presented in Example ??: the pin corresponding to the name for ageneric enumeration of 2ω in ordertype ω. This follows immediately from theclassification of F2-pins in Example 2.3.5 below.


2.3 Classification: general theorems

In this section, we provide a number of general classification theorems for virtualequivalence classes. The theorems are all of the same type: if F is an analyticequivalence relation which is obtained from another equivalence relation E usinga certain operation, then all virtual F -classes are obtained from virtual E-classes using a similar operation. This will provide a suitable background to theinvestigation of specific cases in Section 2.4.

To start with, a great many analytic equivalence relations yield only utterlyuninteresting virtual classes: only those already realized in the ground model.This phenomenon is isolated in the following definition.

Definition 2.3.1. [48, Definition 17.1.2] Let E be an analytic equivalence re-lation on a Polish space X. A virtual E-class represented by 〈P, τ〉 is trivial ifthere is x ∈ X such that P τ E x. The equivalence relation E is pinned if ithas only trivial virtual classes.

The class of pinned equivalence relations has been investigated for a numberof years. The basic pre-existing knowledge about this class is subsumed in thefollowing fact.

Fact 2.3.2. [48, Theorem 17.1.3] The analytic equivalence relations in the fol-lowing classes are pinned:

1. orbit equivalence relations generated by actions of Polish cli groups;

2. Borel equivalence relations with all classes Σ03;

3. equivalence relations Borel reducible to pinned ones.

The operation on equivalence relations which has the most informative transla-tion into virtual classes is that of the Friedman–Stanley jump.

Definition 2.3.3. Let E be an analytic equivalence relation on a Polish spaceX. The Friedman–Stanley jump of E is the equivalence relation E+ on thespace Y = Xω defined by y0 E

+ y1 if [rng(y0)]E = [rng(y1)]E .

Here, the classification of pinned names is right at hand: a pinned name forthe jump is essentially just a set of pinned names for the original equivalencerelation. For a nonempty set S = {〈Pi, τi〉 : i ∈ I} of representatives of virtualE-classes, let τS be the name on the poset QS =

∏i Pi × Coll(ω, I) for an

element of Xω enumerating the set {τi : i ∈ I}.

Theorem 2.3.4. Let E be an analytic equivalence relation on a Polish spaceX.

1. If S is a set of E-pinned names then 〈QS , τS〉 is an E+-pinned name;

2. 〈QS , τS〉 E+ 〈QT , τT 〉 iff the sets S, T represent the same set of virtualE-classes;

2.3. CLASSIFICATION: GENERAL THEOREMS 43

3. whenever 〈P, τ〉 is an E+-pinned name, there is a set S of E-pinned namessuch that 〈P, τ〉 E+ 〈QS , τS〉.

Proof. Items (1) and (2) are immediate. To prove (3), suppose that τ is anE+-pinned name on a poset P . For every virtual E-class y, the statementφ(y) =“rng(τ) contains a realization of the class y” must be decided in thesame way by every condition in P . Let S be any set of E-pinned names whichcollects representatives from all virtual E-classes y such that ∀p ∈ P p φ(y).We claim that the set S works.

Indeed, suppose that G0, G1 ⊂ P are mutually generic filters. The sets[rng(τ/G0)]E and [rng(τ/G1)]E are equal, and by Proposition 2.1.7 they containonly realizations of ground model virtual E-classes. By the choice of the set S,these sets contain exactly realizations of virtual E-classes represented by namesin S. Since the generic ultrafilter G0 was arbitrary, 〈P, τ〉 E+ 〈QS , τS〉 asdesired.

Example 2.3.5. The relation F2 is the Friedman–Stanley jump of the identityon X = 2ω. The identity is pinned by Fact 2.3.2 so its virtual classes can beidentified with elements of X. The F2-classes then correspond to subsets of 2ω.

Countable products of equivalence relations translate to the virtual realm with-out change.

Definition 2.3.6. For each n ∈ ω, let En be an analytic equivalence relationon a Polish space Xn. The product ΠnEn is the equivalence relation E onY =

∏nXn defined by y0 E y1 if for every n ∈ ω, y0(n) En y1(n).

For every function g such that dom(g) = ω and for all n ∈ ω g(n) is someEn-pinned name 〈Qn, τn〉, let τg be the name on the poset Qg =

∏nQn (the

support applied in the product is irrelevant for the equivalence class of theresulting ΠnEn-pin) for the sequence 〈τn : n ∈ ω〉, which is forced to be anelement of Y . The following is nearly immediate.

Theorem 2.3.7. For each n ∈ ω, let En be analytic equivalence relations onrespective Polish spaces Xn and let E =

∏nEn.

1. If g is a sequence of En-pinned names, then 〈Qg, τg〉 is an∏nEn-pin;

2. 〈Qg, τg〉 E 〈Qh, τh〉 iff for each n ∈ ω, g(n) En h(n);

3. whenever 〈P, τ〉 is an E-pinned name, there is a function g such that〈P, τ〉 E 〈Qg, τg〉.

Countable increasing unions of equivalence relations translate to the virtualrealm without change as well.

Theorem 2.3.8. Let {En : n ∈ ω} be an increasing sequence of analytic equiv-alence relations on a Polish space X, and let E =

⋃nEn.


1. Whenever 〈Q, σ〉 is an En-pinned name for some n ∈ ω then it is alsoE-pinned;

2. whenever 〈P, τ〉 is an E-pinned name, there exists a number n ∈ ω andan En-pinned name 〈Q, σ〉 such that 〈P, τ〉 E 〈Q, σ〉.

Proof. (1) is immediate as En ⊂ E. For (2), by the forcing theorem there mustbe conditions p0, p1 ∈ P and a number n such that 〈p0, p1〉 P×P τleft En τright.The transitivity of the relation En then shows that τ on the poset P � p0 is anEn-pinned name. The initial assumptions show that 〈P, τ〉 E 〈P � p0, τ〉. asdesired.

Example 2.3.9. The Louveau jump survives into the virtual realm withoutchange. The Louveau jump of an analytic equivalence relation E on a Polishspace X is the equivalence relation E+L on Y = Xω connecting y0, y1 ∈ Yif for all but finitely many n ∈ ω, y0(n) E y1(n). The Louveau jump canbe written as a countable increasing union of countable products of E, whichby Theorems 2.3.7 and 2.3.8 yields a complete analysis of its virtual classes interms of virtual E-classes. In particular, if E is pinned then its Louveau jumpis pinned.

The virtual realm also correctly reflects the situation in which the equivalenceclasses of one relation consist of countably many equivalence classes of anotherone.

Definition 2.3.10. Let E,F be analytic equivalence relations on a Polish spaceX. We say that F is countable over E if E ⊂ F and every F -class consists ofcountably many E-classes.

Theorem 2.3.11. Let E,F be analytic equivalence relations on a Polish spaceX, with F countable over E.

1. If 〈Q, σ〉 is an E-pinned name then it is F -pinned as well;

2. if 〈P, τ〉 is an F -pinned name then there is an E-pinned name 〈Q, σ〉 suchthat 〈P, τ〉 F 〈Q, σ〉.

Proof. (1) is immediate. For (2), it will be enough to show that there is acondition p ∈ P such that τ is an E-pinned name on P � p, for then 〈P, τ〉 F〈P � p, τ〉 as desired.

Suppose towards a contradiction that there is no condition p ∈ P such thatτ is E-pinned on the poset P � p. It follows by the forcing theorem thatP × P ¬τleft E τright holds. Let M be a countable elementary submodel of alarge structure containing P, τ and the codes for E,F . Use Proposition 1.7.8 tofind an uncountable collection {gi : i ∈ I} of filters on M ∩P pairwise mutuallygeneric over the model M . By the Mostowski absoluteness between the modelsM [gi, gj ] and V for i 6= j ∈ I, the elements τ/gi ∈ X for i ∈ I are pairwiseF -related, but pairwise E-unrelated, contradicting the initial assumptions onthe relations E,F .

2.4. CLASSIFICATION: SPECIFIC EXAMPLES 45

Example 2.3.12. The Clemens jump survives into the virtual realm withoutchange. Here the Clemens jump of an analytic equivalence relation E on a Polishspace X is the equivalence relation E+C on Y = XZ connecting y0, y1 ∈ Y ifthere is n ∈ Z such that for every m ∈ Z y0(m) E y1(m+n). The Clemens jumpis countable over the product of Z-many copies of E, which yields a completeanalysis of its virtual classes by Theorems 2.3.7 and 2.3.8. In particular, if E ispinned, then so is its Clemens jump.

2.4 Classification: specific examples

There are many analytic equivalence relations for which the virtual space canbe classified by more tangible combinatorial objects, but which do not fit intothe context of the theorems of Section 2.3. The purpose of this section is toinvestigate these more difficult, but also more informative, possibilities.

The most interesting issues arise in equivalence relations classifiable by count-able structures. Among these, the Borel equivalence relations are Borel reducibleto an iterated jump of the identity [48, Theorem 12.5.2], so can be handled byTheorems 2.3.4 and 2.3.8. For example, for all equivalence relations E Borelreducible to F2, the virtual E-classes are classifiable by subsets of 2ω by Exam-ple 2.3.5.

More interesting issues arise with analytic equivalence relations. Let E bethe equivalence relation of isomorphism of structures on ω. In this case, thevirtual classes correspond to potential Scott sentences in the sense of [96]. Insome cases, it is possible to classify virtual classes of E by uncountable structuresas in the following definition.

Definition 2.4.1. Let M be a (possibly uncountable) structure of a countablesignature. τM is a Coll(ω,M)-name for some structure on ω isomorphic to M .

It is immediate that the pair 〈Coll(ω,M), τM 〉 is an E-pin, and its E-equivalencerelation does not depend on the choice of the name τM . It turns out that the E-equivalence relation on the E-pins obtained in this way coincides with a familiarconcept of model theory:

Definition 2.4.2. [67, Section 2.4] Let M,N be structures with the same sig-nature. Say that M,N are Ehrenfeucht–Fraisse-equivalent if Player II has awinning strategy in the Ehrenfeucht–Fraisse game. In the EF-game, the twoplayers take turns, at round i ∈ ω Player I starting with an element ni ∈ N ormi ∈ M and Player II responding with an element mi ∈ M or n1 ∈ N respec-tively. After all rounds indexed by i ∈ ω have been completed, Player II wins ifthe map ni 7→ mi for i ∈ ω preserves all relations and functions of N,M in thesignature.

Theorem 2.4.3. Let M,N be models with the same countable signature. Thefollowing are equivalent:

1. M,N are Ehrenfeucht–Fraisse equivalent;


2. 〈Coll(ω,M), τM 〉 E 〈Coll(ω,N), τN 〉.

Proof. For the (1)→(2) direction, if M is Ehrenfeucht–Fraisse equivalent to N aswitnessed by a winning strategy σ for Player I in the EF-game, then Coll(ω,M)×Coll(ω,N) τM E τN , since a generic run of the EF-game in which Player IIfollows the strategy σ will generate an isomorphism between M and N in theextension. For the (2)→(1) direction, suppose that Coll(ω,M) × Coll(ω,N) τM E τN and let π : M → N be a product name for the isomorphism. Thewinning strategy for Player II can be described as follows: as the game develops,Player II also maintains on the side conditions qi ∈ Coll(ω,M)×Coll(ω,N) suchthat q0 ≥ q1 ≥ . . . and qi π(mi) = ni. It is immediate that this is possibleand Player II must win in the end.

In the language of [96], a sentence φ of Lω1ω is grounded if every virtual equiv-alence class is represented by a collapse name for a possibly uncountable modelof φ. In general, there are virtual E-classes which are not represented by astraightforward collapse name as in Definition 2.4.1, as [50, Section 4] shows.However, a collapse name can be found for certain classes of structures.

Definition 2.4.4. Let Γ be a coanalytic set of structures on ω, closed un-der isomorphism. A (possibly uncountable) structure M is a Γ∗∗-structure ifColl(ω,M) τM ∈ Γ.

We proceed to show that for some interesting coanalytic classes Γ, every EΓ-class is represented by a collapse name of a Γ∗∗-structure as in Definition 2.4.1.In the classification results, we always ignore the trivial class of structures whichdo not belong to Γ. Results similar to the following theorem appear in [57] and[65].

Theorem 2.4.5. Let Γ be a coanalytic class of countable structures on ω, in-variant under isomorphism, consisting of rigid structures only.

1. For Γ∗∗-structures, Ehrenfeucht–Fraisse equivalence and isomorphism co-incide;

2. for every EΓ-pin 〈P, σ〉 there is a Γ∗∗-structure M such that 〈P, σ〉 E〈Coll(ω,M), τM 〉.

Proof. Before we begin the argument, note that the statement that every struc-ture in the set A is rigid is Π1

2; therefore, it holds also in all generic extensionsby the Shoenfield absoluteness.

For (1), it is clear that isomorphic structures are Ehrenfeucht–Fraisse equiva-lent. For the opposite implication, suppose that M,N are Γ∗∗ structures whichare EF-equivalent. By Theorem 2.4.3, Coll(ω,M) × Coll(ω,N) τM E τNmust hold. As Γ consists of rigid structures even in the collapse extension,Coll(ω,M)× Coll(ω,N) forces that there is a unique isomorphism π : M → N .Since Coll(ω,M) × Coll(ω,N) is a homogeneous notion of forcing, for eachm ∈ M the value of π(m) is decided by the largest condition to be some

2.4. CLASSIFICATION: SPECIFIC EXAMPLES 47

h(m) ∈ N . The function h : M → N is an isomorphism of M to N presentalready in V .

For (2), let G0×G1 ⊂ P×P be mutually generic filters over V . In the modelV [G0, G1], let N0 = τ/G0 and N1 = τ/G1. To define the model M ∈ V , let x0 ={s : s is the Scott sentence of the model 〈N0, a〉 for some a ∈ N0} ∈ V [G0] andx1 = {s : s is the Scott sentence of the model 〈N1, a〉 for some a ∈ N1} ∈ V [G1].Since N0 is isomorphic to N1, it follows from Karp’s theorem [32, Lemma 12.1.6]that x0 = x1, so x0 = x1 ∈ V [G0] ∩ V [G1] ∈ V . The set x0 will be the universeof the model M . Note that since the model N0 is rigid, the elements of N0 arein one-to-one correspondence with x0 by Karp’s theorem again and the uniqueisomorphism between N0 and N1 factors through the identity on the set x0 = x1.To construct the realizations of relational and functional symbols of the modelM , for every relational symbol R (say binary) of the language of the models ands, t ∈ x0, let s RM t if for the unique a, b ∈ N0 such that s is the Scott sentenceof a and t is a Scott sentence of b, N0 |= s R t. The same definition using themodel N1 yields the same relation, so RM ∈ V [G0] ∩ V [G1] = V . Define therealizations of all functional and relational symbols of the model M in this way.As a result, M is a model in V and the map sending each a ∈ M to the Scottsentence of 〈M,a〉 is an isomorphism of N0 and M in the model V [G0]. Thus,(2) follows.

Theorem 2.4.5 makes it possible to describe some class-sized virtual spaces ex-plicitly:

Example 2.4.6. The virtual Eω1 -classes are precisely classified by ordinals,since Eω1 = EΓ where Γ is the class of all well-orderings on ω. Well-orderingsare rigid, and up to isomorphism are classified by ordinals.

Example 2.4.7. The virtual HC-classes are classified by transitive sets withthe ∈-relation. Note that all extensional, well-founded relations are rigid anduniquely isomorphic to a unique transitive set with the ∈-relation by the Mostowskicollapse theorem [45, Theorem 6.15].

In the case of non-rigid structures, the classification may become more com-plicated. We will treat the case of acyclic graphs, which has the virtue of beingBorel-complete among the equivalence relations classifiable by countable struc-tures. Note that the Ehrenfeucht–Fraisse equivalence on uncountable acyclicgraphs is distinct from isomorphism, as the case of an empty graph on ℵ1 or ℵ2

vertices shows.

Theorem 2.4.8. Let Γ be the class of all acyclic graphs on ω. For every EΓ-pin〈P, σ〉 there is an acyclic graph H such that 〈P, σ〉 E 〈Coll(ω,H), τH〉.

Proof. The point of the proof is that an acyclic graph can be explicitly builtfrom automorphism orbits of its elements. This procedure is captured in thefollowing observation. Suppose x is a set, f : x2 → ω+ 1 is a function such thatf(s, t) > 0↔ f(t, s) > 0, and g : x2 → ω + 1 is a function such that f(s, t) > 0implies g(s) = g(t). Then there is, up to an isomorphism unique, acyclic graph


H(x, f, g) together with an onto map h : y → x, where y is the set of vertices ofH(x, f, g), such that

• for every s, t ∈ x and every vertex v ∈ y, if h(v) = s then the set of allneighbors of v mapped to t has size f(s, t);

• for every s ∈ x there are g(s) many connected components of the graphH containing a vertex v with h(v) = s.

The construction of the graph H(x, f, g) is straightforward. Note that when-ever u, v ∈ y are two vertices such that h(u) = h(v) then there is an automor-phism of the graph H(x, f, g) sending u to v.

Now, suppose that σ is an E-pinned name on a poset P and let G0 ×G1 ⊂P×P be mutually generic filters over V . In the model V [G0, G1], let H0 = σ/G0

and H1 = σ/G1. To define the graph H ∈ V , let x0 = {s : s is the Scott sentenceof the model 〈H0, v〉 for some vertex v of H0} ∈ V [G0] and x1 = {s : s is theScott sentence of the model 〈H1, v〉 for some vertex v in H1} ∈ V [G1]. SinceH0 is isomorphic to H1, it follows from Karp’s theorem [32, Lemma 12.1.6]that x0 = x1, so x = x0 = x1 ∈ V [G0] ∩ V [G1] ∈ V . Let f : x2 → ω + 1be the function defined by f(s, t) = i if every vertex of H0 of type s has i-many neighbors of type t when i ∈ ω, and f(s, t) = ω if every vertex of H0

of type s has infinitely many neighbors of type t. Let g : x → ω + 1 be thefunction defined by f(s) = i if there are i-many connected components of H0

containing a node of type s when i ∈ ω, and g(s) = ω if there are infinitelymany connected components of H0 containing a node of type s. Note that thesefunctions are well-defined and the graph H0 is isomorphic to H(x, f, g) in themodel V [G0]. Similar definitions using the grap H1 yield the same functions, sof, g ∈ V [G0] ∩ V [G1] = V . Working in V , consider the graph H = H(x, f, g).This graph is isomorphic to H0 in V [G0], so τH E σ.

In the recent years, model theorists proved several other theorems regarding theclassification of virtual isomorphism classes by possibly uncountable structures.[96] provides further examples of grounded sentences beyond those obtained byTheorems 2.4.5 and 2.4.8. The first item of the following theorem is related toclassical arguments going at least as far back as the unpublished work of LeoHarrington in the 1980’s. Similar results also appear in [6] and [65].

Fact 2.4.9. [50, Section 4] Let E be the equivalence relation of isomorphism ofstructures on ω.

1. If P is a poset forcing |ℵV2 | > ℵ0 then every E-pin 〈P, σ〉 is E-equivalentto 〈Coll(ω,M), τM 〉 for some structure M of size ≤ ℵ1;

2. if P forces |ℵV2 | = ℵ0, then there is an E-pin 〈P, σ〉 which is not E-equivalent to 〈Coll(ω,M), τM 〉 for any structure M .

Virtual spaces for equivalence relations which are not classifiable by countablestructures are often quite difficult to understand. To conclude this section, westate another classification theorem and a couple of open questions.

2.5. CARDINAL INVARIANTS 49

Theorem 2.4.10. Let E be the equivalence relation on X = (P(ω))ω connectingx0, x1 if rng(x0) and rng(x1) generate the same filter on ω. The virtual E-classesare classified by filters on ω.

Proof. On one hand, if F is a filter on ω, one can consider the Coll(ω, F )-nameτF for a generic enumeration of the filter F . It is immediate that the name τFis E-pinned, and distinct filters yield inequivalent names.

For the more difficult part, suppose that P is a partial order and τ is anE-pinned name on P ; we must find a filter F on ω such that τ is equivalentto τF . To do this, let F = {a ⊂ ω : ∃p p a belongs to the filter generatedby rng(τ)}. By the pinned property of the name τ , the existential quantifier inthe definition of F can be replaced by universal without changing the resultingset F . It follows immediately that the set F is a filter; we must show that τ isequivalent to τF .

Suppose towards a contradiction that this fails; then it must be the case thatfor some condition p ∈ P and some number n ∈ ω, p forces τ(n) to have nosubset in the filter F . Since the name τ is E-pinned, there must exist conditionsp0, p1 ≤ p and a finite set a ⊂ ω such that 〈p0, p1〉

⋂m∈a τright(m) ⊂ τleft(n).

Let b = {k ∈ ω : ∃r ≤ p1 r k ∈⋂m∈a τ(m)} and let c = {k ∈ ω : p0

k ∈ τ(n)}. Observe that b ∈ F (since p1 ⋂m∈a τ(m) ⊂ b) and c /∈ F (since

p0 c ⊂ τ). Thus, there has to be a number k ∈ b \ c, and for this number kthere are conditions r0 ≤ p0 and r1 ≤ p1 such that r1 k ∈

⋂m∈a τ(m) and

r0 k /∈ τ(n). This, however, contradicts the choice of the conditions p0, p1.

Consider the equivalence relation E of homeomorphism of compact metrizablespaces. This is known to be the largest equivalence relation reducible to an orbitequivalence in the sense of Borel reducibility [105]. In an attempt to describe itsvirtual space, consider any compact Hausdorff space X with a topology basis ofsize κ, and consider the Coll(ω, κ)-name τX for the interpretation of the spaceX in the extension in the sense of [102]. The interpretation of X will have acountable basis, so the interpretation will be a compact metrizable space. Thebasic theory of interpretations shows that 〈Coll(ω, κ), τX〉 is an E-pin. The mostnatural question is open:

Question 2.4.11. Is every virtual E-class represented by a compact Hausdorffspace?

The measure equivalence E is one of the hardest equivalence relations to under-stand. It connects two Borel probability measures µ, ν on the Cantor space ifthey share the same ideal of null sets. The relation F2 Borel reduces to E, soE is not pinned.

Question 2.4.12. Classify the virtual space for the measure equivalence.

2.5 Cardinal invariants

There are several cardinal invariants of equivalence relations which are associ-ated with the concept of virtual equivalence classes. They respect the Borel


reducibility order; therefore, they are useful as tools for nonreducibility results.They also are conceptually useful in several places in this book.

2.5a Basic definitions

The following definition records the most natural cardinal invariants associatedwith the virtual spaces.

Definition 2.5.1. Let E be an analytic equivalence relation on a Polish spaceX.

1. κ(E), the pinned cardinal of E, is the smallest κ (if it exists) such thatevery virtual E-class has a representative on a poset of size < κ. If E ispinned, we let κ(E) = ℵ1. If κ(E) does not exist, we write κ(E) =∞;

2. λ(E) is the cardinality of the set of all virtual E-classes. If the virtualE-classes form a proper class, we let λ(E) =∞;

3. λ(E,P ) is the cardinality of the set of all virtual E-classes represented onthe poset P .

Note that κ(E) = ∞ just in case λ(E) = ∞; at the same time, λ(E,P ) < ∞holds for all E,P . The rather mysterious demand that κ(E) = ℵ1 for all pinnedequivalence relations E is explained by a reference to Theorem 2.6.2: there areno nontrivial pinned names on countable posets for any analytic equivalencerelation. It turns out that the cardinals κ(E) and λ(E,P ) can attain all kindsof exotic and informative values. We will start with the two archetypal andsomewhat boring computations.

Example 2.5.2. κ(F2) = c+ and λ(F2) = 2c.

Proof. This follows immediately from the classification of pinned names for theFriedman–Stanley jump. Every virtual F2 class is represented by a subset of2ω, and distinct subsets of 2ω give rise to distinct virtual F2 classes.

Example 2.5.3. κ(Eω1) = ∞. To see this, note that by Example 2.4.6, thevirtual Eω1-classes are classified by ordinals, so there is a proper class of them.Similarly, κ(HC) =∞, as by Example 2.4.7 the virtual HC-classes are classifiedby transitive sets, so there is a proper class of them.

Theorem 2.8.9 shows that in fact Eω1 is a minimal example of an equivalencerelation E with κ(E) =∞.

The most appealing fact about the cardinal invariants κ(E) and λ(E,P )is that they respect the Borel reducibility order; therefore, they can be usedto prove Borel nonreducibility results. In a good number of instances, thecomparison of the cardinal invariants is the fastest and most intuitive way ofproving nonreducibility. One of the main features of this style of argumentationis that it automatically survives the transfer to nonreducibility by functionsmore complicated than Borel.


Theorem 2.5.4. Let E,F be analytic equivalence relations on Polish spacesX,Y respectively. If E ≤a F then κ(E) ≤ κ(F ) and λ(E,P ) ≤ λ(F, P ) holdsfor every partial order P .

Proof. Suppose that h : X → Y is a Borel function witnessing the reduction ofE to F everywhere except for a set Z ⊂ X consisting of countably many E-classes. By a Shoenfield absoluteness argument, these properties of the functionh transfer to all generic extensions. If P is a partial ordering and τ is an E-pinned name on P which is not a name for one of the classes in the set Z, thenh(τ) is an F -pinned name on P , and the map τ 7→ h(τ) respects virtual E- andF -classes, and it is an injection from the former to the latter. It follows thatλ(E,P ) ≤ λ(F, P ).

Now, suppose that Q is another poset and that h(τ) has a F -equivalent nameσ on Q. By the Shoenfield absoluteness, Q ∃x ∈ X \Z h(x) = τ ; any Q-namefor such an x is E-pinned and the name for h(x) is F -equivalent to h(τ). Abrief bout of diagram chasing now shows that κ(E) ≤ κ(F ) and the theoremfollows.

The following neat application has been observed by [96] for the case of Borelequivalence relations classified by countable structures.

Example 2.5.5. Friedman and Stanley proved that the Friedman–Stanleyjump of a Borel equivalence relation E is not Borel reducible to E [31]. The mostappealing proof of this fact uses the cardinal invariant λ(E). Theorem 2.5.6 be-low shows that for Borel equivalence relations, the value of λ(E) is an actualcardinal as opposed to∞. Theorem 2.3.4 shows that virtual E+-classes are clas-sified by sets of virtual E-classes, in other words λ(E+) = 2λ(E). Theorem 2.5.4then completes the argument.

2.5b Estimates

The key fact about the pinned cardinal is the following theorem. It shows thatthere are a priori bounds on the size of the pinned cardinal; in particular, if theequivalence relation E is Borel, then κ(E) ≤ iω1 and the virtual space of E isa set. Recall that the i function is defined by recursion: i0 = ℵ0, iα+1 = 2iα ,and iα = supβ∈α iβ if α is a limit ordinal.

Theorem 2.5.6. Let E be a Borel equivalence relation on a Polish space X ofrank Π0

α. Then κ(E) ≤ (iα)+.

Proof. Let τ be an E-pinned name on a poset P ; we must produce a Coll(ω,iα)-name σ which is E-related to τ . Note that [τ ]E is a P -name for a Borel setof rank ≤ α. As is the case for every name for a Borel set, [87, Corollary2.9] shows that in the Coll(ω,iα) extension V [G] there is a Borel code for aBorel set B ⊂ X such that in every further forcing extension V [G][H] and everyx ∈ X ∩ V [G][H] in that extension, x ∈ B if and only if V [x] |= P x ∈ [τ ]E .Note that if H ⊂ P is generic over V [G], then the set B is nonempty in V [G][H],containing the point τ/H; this follows from the fact that τ is E-pinned. Thus,


the set B is nonempty already in V [G] by the Mostowski absoluteness betweenV [G] and V [G][H]. Back in V , let σ be any Coll(ω,iα)-name for an element ofthe set B. This clearly works.

Example 2.5.7. For any given countable ordinal α, let Γα be the class of binaryrelations on ω which are extensional and wellfounded of rank < α; let Eα bethe isomorphism relation. It is not difficult to check (Theorem 2.4.5) that foreach countable ordinal α, the relation Eα is Borel, and its pinned names arecollapse names for isomorphs of the membership relation on sets in Vα. Thus,the cardinals κ(Eα) converge to iω1

.

Theorem 2.5.8. Let E be an analytic equivalence relation almost reducible toan orbit equivalence relation of a continuous Polish group action. If κ(E) <∞then κ(E) is not greater than the first ω1-Erdos cardinal.

Proof. Let κ be the first ω1-Erdos cardinal, and suppose that κ(E) > κ; wemust show that κ(E) = ∞. Since the cardinal κ is the Hanf number for theclass of wellfounded models of first order sentences, for every cardinal λ thereis a wellfounded model M such that M |= κ(E) > λ. Now, since E is almostreducible to an orbit equivalence relation, Corollary 2.7.4 shows that the well-founded model M is correct about κ(E) to the extent that |κ(E)M | ≤ κ(E). Itfollows that κ(E) > λ, and since λ was arbitrary, κ(E) =∞.

Example 2.5.9. For every countable ordinal α there is a coanalytic class Γ ofstructures on ω, invariant under isomorphism, such that κ(EΓ) is equal to thefirst α-Erdos cardinal.

Proof. Let Γ be the class of all binary relations on ω which are extensional,wellfounded, and do not admit a sequence of indiscernibles of ordertype α; weclaim that this class works.

Clearly, Γ is a coanalytic set of rigid structures invariant under isomorphism.Write E for EΓ and κ for the first α-Erdos cardinal. By Theorem 2.4.5, everyvirtual E-class is represented by a transitive set A without indiscernibles ofordertype α. It must be the case that |A| < κ so κ(E) ≤ κ. On the other hand,whenever λ < κ is an ordinal, then the structure 〈Vλ,∈〉 has no indiscernibles ofordertype α, and it remains such in every forcing extension by a wellfoundednessargument. Thus, the Coll(ω, Vλ)-name for the generic isomorph of this structureis E-pinned, and it is not equivalent to any E-pinned name on a poset of size< |Vλ| since it entails the collapse of |Vλ| to ℵ0. Thus, κ(E) = κ as desired.

Theorem 2.5.10. Let E be an analytic equivalence relation on a Polish spaceX. Let κ be a measurable cardinal. If κ(E) <∞ then κ(E) < κ.

Proof. Suppose that there is a poset P and an E-pinned name τ on P which isnot E-related to any name on a poset of size < κ. We will produce a properclass of pairwise non-E-related E-pinned names.

First note that the poset P and the name τ can be selected so that |P | = κ.Simply take an elementary submodel M of size κ of large structure with Vκ ⊂M


and consider Q = P ∩M and σ = τ ∩M ; so |Q| = κ. As M is correct aboutpinned names and the equivalence E by a Shoenfield absoluteness argument, σis an E-pinned name on Q and it is not E-equivalent to any pinned names onposets of size < κ.

Thus, assume that the poset P has size κ. Let j : V → N be any elementaryembedding into a transitive model with critical point equal to κ. Note thatH(κ) ⊂ N , so both P, τ are (isomorphic to) elements of N . Let 〈Nα, jβα : β ∈ α〉be the usual system of iteration of the elementary embedding j along the ordinalaxis. Let Pα = j0α(P ) and τα = j0,α(τ). It will be enough to show that the pairs〈Pα, τα〉 for α ∈ Ord are pairwise E-unrelated. To see this, pick ordinals α ∈ β.As the original poset had size κ, it is the case that Pα, τα, Pβ , τβ are in the modelNβ . By the elementarity of the embedding j0β , Nβ |= ¬〈Pα, τα〉 E 〈Pβ , τβ〉. Thewellfounded modelNβ is correct about E by a Shoenfield absoluteness argument,so 〈Pα, τα〉 E 〈Pβ , τβ〉 fails also in V as required.

Unlike the previous theorems in this section, we do not have a complementaryexample showing that the measurable cardinal bound is, at least to some extent,optimal.

The last theorem in this section provides an estimate of the λ cardinal forunpinned equivalence relations. The well-known Silver dichotomy says thatevery Borel equivalence relation with uncountably many classes has in fact 2ℵ0

many classes. We would like to show that every unpinned Borel equivalencerelation has at least 2ℵ1 many virtual classes. However, in ZFC this is stillopen. The best we can do is the following.

Theorem 2.5.11. Let E be an unpinned Borel equivalence relation. Let κ bean inaccessible cardinal. Then Coll(ω,< κ) λ(E,Coll(ω, ω1)) = 2ℵ1 .

Proof. Let A ⊂ P(κ) be a set of size 2κ such that all elements of A havecardinality κ and any two distinct elements of A have intersection of cardinality< κ. Recall that the poset P = Coll(ω,< κ) is a finite support product of posetsColl(ω, α) for α ∈ κ. For every set a ⊂ κ write Pa = Πα∈aColl(ω, α) ⊂ P . LetG ⊂ Coll(ω,< κ) be a generic filter. Use Theorem 2.7.1 to argue that for everya ∈ A, V [G∩Pa] |= E is unpinned, and let 〈Qa, τa〉 ∈ V [G∩Pa] be a nontrivialE-pinned name with its attendant poset. It will be enough to show that thepairs 〈Qa, τa〉 for a ∈ A represent pairwise distinct virtual classes.

Suppose towards a contradiction that for some a 6= b in the set A the E-pinned names τa, τb are equivalent. Let c = a∩b and work in the model V [G∩Pc].The assumptions imply that the poset (Pa\c ∗ Qa)× (Pb\c ∗ Qb) forces (at leastbelow some condition) that τa E τb. It follows that the name τa on the posetPa\c ∗ Qa is E-pinned. Since the equivalence relation E is Borel, it follows that

that the virtual class of 〈Pa\c ∗ Qa, τa〉 is also represented by some pair 〈R, σ〉by some poset of size < iω1

< κ by Theorem 2.5.6. However, in the modelV [G ∩ Pa], there is a filter H ⊂ R generic over the model V [G ∩ Pc]. Thus,it would have to be the case that in the model V [G ∩ Pa], Qa τa E σ/H,contradicting the nontriviality of the name τa.


Example 2.5.12. As with the Silver dichotomy, the conclusion of Theorem 2.5.11fails for analytic equivalence relations. Consider the case of the equivalence re-lation Eω1

and an inaccessible cardinal κ such that 2κ > κ+. In the Coll(ω,<κ) extension, ℵ2 < 2ℵ1 will hold; at the same time, Eω1

-names realized onColl(ω, ω1) are classified by ordinals of cardinality ℵ1, so there are only ℵ2-many of them.

2.5c Cardinal arithmetic examples

The cardinal invariants κ(E) and λ(E) provide a basis on which equivalence re-lations can be compared with uncountable cardinals of all sorts. In this section,we will introduce several jump operations which have direct translations intocardinal arithmetic operations. Using this approach, one can formally encodestatements such as the failure of the singular cardinal hypothesis into reducibil-ity results between Borel or analytic equivalence relations. For brevity, givena coanalytic class Γ of structures on ω invariant under isomorphism, we writeκ(Γ) for κ(EΓ) in this section.

The constructions in this section depend on certain types of jump operatorson structures and equivalence relation. They are all provisionally denoted bythe + sign, not to be confused with the Friedman–Stanley jump. The first jumpoperation on equivalence relations we will consider translates into the powersetoperation using the pinned cardinal:

Definition 2.5.13. Let Γ be a coanalytic class of structures on ω, invariantunder isomorphism. Γ+ is the class of structures on ω of the following descrip-tion: there is a partition ω = a ∪ b into two infinite sets, there is a Γ-structureon a, and there is an extra relation R on b × a such that the vertical sectionsRm for m ∈ b are pairwise distinct subsets of a.

Proposition 2.5.14. Let Γ be a coanalytic class of structures on ω, invariantunder isomorphism.

1. Γ+ is coanalytic, and if Γ is Borel then so is Γ+;

2. if Γ consists of rigid structures, then so does Γ+;

3. if Γ consists of rigid structures and κ(Γ) = λ+ then κ(Γ+) = (2λ)+.

Proof. The first two items are obvious. For (3), suppose that κ(Γ) = λ+. Thenthere must be a Γ∗∗ structure M of size λ and no Γ∗∗ structures of largersize. By Theorem 2.4.5 and (2), every virtual EΓ-class is associated with a Γ∗∗

structure. Every (Γ+)∗∗-structure consists of a Γ∗∗-structure and some family ofits pairwise distict subsets; the largest such structure then is of size 2λ exactly.This completes the proof.

Corollary 2.5.15. For every countable ordinal α there is a Borel class Γ ofrigid structures such that κ(EΓ) = i+

α .


Proof. By transfinite recursion on α define Borel classes Γα consisting of rigidstructures as follows. Let Γ0 be the class of structures isomorphic to 〈ω,∈〉. LetΓα+1 = (Γα)+. For a limit cardinal α let Γα be the class of structures whichconsist of exactly one copy of a structure in class Γβ for each β ∈ α. It is notdifficult to prove by induction on α using the Proposition 2.5.14 at the successorstage and Theorem 2.4.5 at the limit stage that κ(EΓ) = i+

α as desired.

Definition 2.5.16. Let Γ be a coanalytic class consisting of structures on ωinvariant under isomorphism. Γ+ is the class of structures on ω of the followingdescription. A structure M ∈ Γ+ has a linear order ≤M . Moreover, writing Lnfor the set {m ∈ M : m ≤M n}, the structure M induces a Γ-structure Mn onLn for coboundedly many n ∈ M such that for distinct elements n0, n1 ∈ M ,the structures 〈Mn0 ,≤M 〉 and 〈Mn1 ,≤M 〉 are nonisomorphic.

Proposition 2.5.17. Let Γ be a coanalytic class of structures on ω, invariantunder isomorphism.

1. Γ+ is coanalytic;

2. if Γ is Borel and consists of rigid structures, then Γ+ is Borel and consistsof rigid structures;

3. if Γ consists of rigid structures, then κ(Γ+) = κ(Γ)+.

Proof. (1) is clear. For (2), the rigidity conclusion is clear. To see the Borelnessof the class Γ+ note that the statement “〈M,≤M 〉 is isomorphic to 〈N,≤N 〉”for structures M,N ∈ Γ is Borel by the rigidity of the structures in Γ and theLusin–Suslin theorem [55, Theorem 15.1]. For (3), write κ = κ(Γ). To showthat κ(Γ+) ≤ κ+, use Theorem 2.4.5 to argue that every virtual EΓ+ class isclassified by (Γ+)∗∗-structure. Such a structure contains a linear ordering, andon cofinally many initial segments of the ordering there is a Γ∗∗-structure. Itfollows that every initial segment of the ordering is of cardinality less than κ,so the underlying set has size at most κ.

To show that κ(Γ+) ≥ κ+, treat first the case that κ is a limit cardinal. LetM be the structure on κ which includes the usual ∈-ordering on κ and on eachcardinal λ < κ on which there is a Γ∗∗-structure, M contains one. This is a(Γ+)∗∗-structure of size κ, so κ(Γ+) ≥ κ+.

Suppose now that κ is a successor cardinal, κ = λ+. This means that thereis a Γ∗∗-structure N of size λ. Let M be the structure on κ which includes theusual ∈-ordering of κ and on each ordinal µ < κ of cardinality λ, it contains acopy of N . This is a (Γ+)∗∗-structure of size κ, so κ(Γ+) ≥ κ+ in this case aswell. The proof is complete.

Corollary 2.5.18. For every countable ordinal α > 0 there is a Borel equiva-lence relation E such that (provably) κ(E) = ℵα.

Proof. Now, by recursion on a countable ordinal α construct a Borel class Γαof rigid structures as follows. Γ0 is the set of structures isomorphic to 〈ω,∈〉,


containing just one isomorphism class. Then let Γα+1 = Γ+α and Γα =

⋃β∈α Γβ

if α is limit. Proposition 2.5.17 can be used to show by transfinite inductionthat κ(Γα) = ℵ1+α as desired.

Note that an equivalence relation E as above for α ≥ 2 cannot be reducibleto F2 and F2 cannot be reducible to it. This answers a question of Kechris[48, Question 17.6.1] in the negative as well as some related questions of SimonThomas. To see that F2 cannot be Borel reducible to any E, suppose for con-tradiction that h : dom(E) → dom(F2) is a Borel reduction. Pass to a genericextension in which c > ℵω1

. There, h is still a reduction of E to F , whileκ(E) > κ(F ). This contradicts Theorem 2.5.4. To see that E cannot be re-ducible to F2 for any α > 2, pass to a generic extension in which the ContinuumHypothesis holds instead.

The following two examples deal with jump operations designed to mimiccardinal exponentiation. They lead to nonreducibility results which, at least onthe face of it, use the failure of singular cardinal hypothesis in various situations.This means that the proofs presented use large cardinal assumptions, as they areneeded to get the failure of the singular cardinal hypothesis. We make no claimas to whether the large cardinal assumptions are necessary for the conclusion.

Definition 2.5.19. Let Γ,∆ be coanalytic classes of structures on ω, invariantunder isomorphism. The symbol Γ∆ stands for the coanalytic class of structuresM on ω of the following form: ω is partitioned into infinite sets ω = a ∪ b ∪ c,on M � a is a structure in class Γ, M � b is a structure in class ∆, and there isan extra relation R ⊂ c × b × a such that for every m ∈ c, the vertical sectionRm is a function from b to a, and for m0 6= m1 ∈ c the vertical sections Rm0

and Rm1are distinct.

Proposition 2.5.20. Suppose that the classes Γ, ∆ consist of rigid structuresonly. Then

1. Γ∆ consists of rigid structures only;

2. κ(Γ∆) = sup{κλ : κ < κ(Γ), λ < κ(∆)}.

Proof. The first item is nearly trivial. For the second item, use Theorem 2.4.5 tonote that a (Γ∆)∗∗ structure is represented by a Γ∗∗-structure, a ∆∗∗-structure,and an infinite set of functions from the latter to the former.

Example 2.5.21. Let ∆ be the class of structures isomorphic to 〈ω,∈〉 (con-sisting of just one equivalence class). For every countable ordinal α, let Γα bethe class derived in Corollary 2.5.18. Let E be the equivalence relation of iso-morphism on the class (Γω)∆ and F be the isomorphism equivalence relationon the class EΓα . Then E is not Borel reducible to F2 × F .

Proof. Move to a model of ZFC where c = ℵ1 and ℵℵ0ω > ℵα. Proposition 2.5.20shows that κ(E) = ℵℵ0ω > c · ℵα = κ(F2 × F ). The conclusion of the examplefollows from Theorem 2.5.4.


Definition 2.5.22. Let E be an analytic equivalence relation on a Polish spaceX and F be a Borel equivalence relation on a Polish space Y . EF is theequivalence relation (F ×E)+ on the space (Y ×X)ω restricted to the Borel set{z ∈ (Y ×X)ω : rng(z) is a partial function from Y to X whose domain consistsof pairwise F -unrelated elements}.

It seems to be impossible to formulate this concept in an analytic form withoutthe additional demand that F be Borel.

Proposition 2.5.23. Let E be an analytic equivalence relation on a Polishspace X and F be a Borel equivalence relation on a Polish space Y . Then

1. λ(EF ) = λ(E)λ(F );

2. if P is a partial order such that P |λ(F )| = |κ(F )| = ℵ0 then λ(EF , P ) =λ(E,P )λ(F ).

Note that as F is assumed to be Borel, the value of λ(F ) is not ∞.

Proof. Both statements follow from the classification of virtual classes for prod-uct and Friedman–Stanley jump in Theorem 2.3.4. An EF -virtual class is rep-resented by a function from F -virtual classes to E-virtual classes.

Example 2.5.24. Let F0 be any Borel equivalence relation with countablymany classes and let F1 be the identity on 2ω. For every Borel equivalencerelation E, EF1

ω1is not Borel reducible to E × EF0

ω1.

Proof. Move to a model of set theory where c = ℵ1 and there is a cardinal κ >iω1

such that (κ+)ℵ1 > (κ+)ℵ0 . The first such a cardinal must be in violationof the singular cardinal hypothesis at cofinality ω1. Let P = Coll(ω, κ). Clearly,λ(Eω1 , P ) = κ+, since the pinned names on P correspond to ordinals below κ+.Since E is Borel, λ(E), κ(E) < iω1 < κ by Theorem 2.5.6, so Proposition 2.5.23shows that λ(E × EF0

ω1) = λ(E) · (κ+)ℵ0 = (κ+)ℵ0 < (κ+)ℵ1 = λ(EF1

ω1). The

argument is concluded by a reference to Theorem 2.5.4.

A similar argument with a failure of the singular cardinal hypothesis at largercofinalities yields

Example 2.5.25. Let F be any Borel equivalence relation. For every Borelequivalence relation E, EF+

ω1is not Borel reducible to E × EFω1

.

2.5d Hypergraph examples

The previous examples were to some extent artificial in the sense that the valuesof the pinned cardinal were directly built into them. The following examples, allBorel reducible to F2, are connected with combinatorics of small uncountablecardinals via the pinned cardinal, even though the connection is not at firstsight obvious.


Definition 2.5.26. A hypergraph on a set X is a subset of X≤ω. If G is ahypergraph on X, a G-anticlique is a set A ⊂ X such that A≤ω ∩ G = 0. IfG is an analytic hypergraph on a Polish space X, write EG for the equivalencerelation on Xω connecting y, z if the sets rng(y), rng(z) either both fail to beG-anticliques or they are equal.

It is clear that the equivalence relations of the form EG are all analytic andalmost Borel reducible to F2. In the common case when G is Borel and containsonly finite edges, the equivalence relation is in fact Borel. The following propo-sition provides a simple combinatorial characterization of the pinned cardinalof equivalence relations of this form.

Proposition 2.5.27. Let G be an analytic hypergraph on a Polish space X.Then κ(EG) is equal to the larger of ℵ1 and the minimal cardinal κ such thatthere is no G-anticlique of size κ.

Proof. Write E = EG and write κ for the minimum cardinality in which there isno G-anticlique. To exclude trivial cases, assume that κ is uncountable. For the≤-inequality in the proposition, let Q be a poset and σ be an E-pinned Q-name.By Example 2.3.5, there must be a set A ⊂ X such that Q rng(σ) = A. If A isnot an anticlique then the E-pin 〈Q, σ〉 is trivial as it is forced to belong to thesingle E-class consisting of enumerations of sets which are not anticliques. If Ais an anticlique, then the E-pin 〈Q, σ〉 is equivalent to a pin on the Coll(ω, |A|)-poset which has size less than κ.

For the ≥-inequality, suppose that A ⊂ X is an anticlique. A simple well-foundedness argument shows that A remains an anticlique in any generic exten-sion. Thus, the Coll(ω,A)-name τA for a generic enumeration of A is E-pinnedan the E-pin 〈Coll(ω,A), τA〉 can be equivalent only to E-pins on posets whichmake the cardinality of A countable.

The values of cardinals defined in this way are subject to forcing manipulationsand transfinite combinatorics. The interesting examples are always connectedwith a universality feature of the hypergraph in question. Our first two examplesuse Borel hypergraphs of finite arity.

Definition 2.5.28. Let X be a Polish space. A Borel relation R ⊂ [X]<ℵ0 ×Xis combinatorially universal if it has countable vertical sections and for everycardinal κ and every relation T ⊂ κ<ℵ0 × κ with countable vertical sections,there is a c.c.c. poset P adding an injective function π : κ → X which is ahomomorphism of T to R.

It is not clear whether combinatorial universality of this sort is actually a prop-erty absolute among transitive models of ZFC, but all universal examples foundin this section are absolutely universal.

Theorem 2.5.29. Let X be a Polish space. There is a combinatorially universalBorel relation on [X]<ℵ0 ×X with countable vertical sections.


Proof. Let X = P(ω). We will show that the relation R ⊂ [X]<ℵ0 ×X definedby 〈a, x〉 ∈ R if x is computable from a is universal. Let κ be a cardinal and letT ⊂ [κ]n × κ be a relation with countable vertical sections. Let 〈km : m ∈ ω〉be a recursive sequence of increasing functions in ωω with disjoint ranges. Fora finite set b ⊂ P(ω) let eb be the increasing enumeration of the set

⋂b, for

every m ∈ ω let hm(b) ⊂ P(ω) be the set of all l ∈ ω such that eb ◦ km(l) is anodd number. We will produce a forcing which adds an injection π : κ→ X suchthat for every finite set a ⊂ κ and every α ∈ Ta, there is a number m ∈ ω suchthat π(α) is modulo finite equal to hm(π′′a).

Let P be the poset of all tuples p = 〈np, πp, νp〉 so that

• np ∈ ω, πp is a partial function from κ to P(np) with finite domain dom(p);

• νp is a finite partial function from P(dom(p)) × ω to dom(p) such that(a, νp(a,m)) ∈ T whenever 〈a,m〉 ∈ dom(νp).

The ordering on P is defined by q ≤ p if np ≤ nq, dom(p) ⊂ dom(q),∀α ∈ dom(p) πp(α) = πq(α) ∩ np, νp ⊂ νq, and for every 〈a,m〉 ∈ dom(νp),whenever l is a number in the domain of (eπ′′q a \ eπ′′q a) ◦ km then eπ′′q a ◦ km(l) isodd if and only if l ∈ πq(νp(a,m)). It is not difficult to see that P is indeed anordering.

Claim 2.5.30. The poset P is c.c.c.

Proof. Let 〈pα : α ∈ ω1〉 be conditions in P . The usual ∆-system and countingarguments can be used to thin down the collection if necessary so that the setsdom(pα) for α ∈ ω1 form a ∆-system with root b and for all a ∈ [b]n andall α ∈ ω1, Ta ∩ dom(pα) ⊂ b. Moreover, we can require that the increasingbijection between dom(pα) and dom(pβ) extends to an isomorphism of pα andpβ for every α, β ∈ ω1.

We claim that any two conditions in such a collection are compatible. Indeed,whenever α, β ∈ ω1, then the condition q defined by nq = nqα , πq = πpα∪πpβ andνq = νpα ∪ νpβ is easily checked to be a common lower bound of the conditionspα, pβ .

Claim 2.5.31. Whenever a ∈ [κ]n and β ∈ Ta, the set Da,β = {p ∈ P : a∪{β} ⊂dom(p),∃m νp(a,m) = β} is dense in P .

Proof. Let p ∈ P ; we must find a condition q ≤ p in the set Da,β . For defi-niteness assume that β /∈ dom(p). Choose m ∈ ω such that 〈a,m〉 /∈ dom(νp).Consider the condition q ≤ p defined by nq = np, πq = πp ∪ {〈α, 0〉 : α ∈a \ dom(p), 〈β, 0〉}, νq = νp ∪{〈a,m, β〉}. The condition q ≤ p is in the set Da,β

as required.

Claim 2.5.32. For every finite set a ⊂ κ and every k ∈ ω, the set Da,k = {p ∈P : a ⊂ dom(p) and the set

⋂π′′pa has at least k elements} is dense in P .


Proof. Fix a, k and let p ∈ P be an arbitrary condition. We must find a condi-tion q ≤ p in the set Da,k. First of all, the previous claim shows that one canstrengthen p to include all ordinals in a. Increasing np if necessary, we may alsoassume that k < np.

Consider the set b = π′′pa and the function eb; write k′ = dom(eb). If k ≤ k′then q = p will work. Otherwise, it is easy to find an increasing sequenced = 〈mi : k

′ ≤ i < k〉 of numbers larger than np such that, writing e = eb ∪ d,for every natural number m such that 〈a,m〉 ∈ dom(νp) and every l such thatk′ ≤ km(l) < k, mkm(l) is odd if and only if l ∈ p(νr(a,m)). The conditionq ≤ p defined by nq = mk−1 + 1, dom(πq) = dom(πp), ∀β ∈ a πq(β) = πp(a) ∪{mi : k

′ ≤ i < k}, ∀β ∈ dom(πp) \ a πq(β) = πp(β), and νq = νp, is in the setDa,k as desired.

The last two claims show that the function π : κ → X defined as π(α) =⋃{πp : p is in the generic filter} is forced by P to be the desired homomorphism.

Recall that if R ⊂ [X]ℵ0×X is a relation then a ⊂ X is R-free if for every x ∈ a〈a \ {x}, x〉 /∈ R.

Example 2.5.33. Let X be a Polish space and let R ⊂ [X]<ℵ0 × X be acombinatorially universal Borel relation with countable vertical sections. Letn ≥ 1 be a number. Let Gn be the hypergraph of all sets a ∈ [X]n which areR-free. Then

1. κ(EGn) ≤ ℵn+1;

2. if Martin’s Axiom for ℵn holds then equality is attained.

Proof. Fix the number n ∈ ω. The argument depends on an old theorem ofSierpinski [35]: ℵn+1 is the smallest cardinal κ such that every relation T ⊂[κ]n × κ with countable vertical sections has a T -free n + 1-tuple. To prove(1), apply the Sierpinski theorem to show that there is no Gn-anticlique of size≥ ℵn+1. To prove (2), let κ = ℵn and use the Sierpinski theorem again tofind a relation T ⊂ κ<ℵ0 × κ with countable vertical sections and no T -free n-tuple. Then use the universality assumption to find an injective homomorphismπ : κ → X which is a homomorphism of T to R. Observe that rng(π) is aGn-anticlique of size κ. Proposition 2.5.27 then implies that κ(EGn) ≥ κ+ =ℵn+1.

Definition 2.5.34. Let X be a Polish space. A Borel equivalence relation Ron [X]<ℵ0 with countably many classes is combinatorially universal if for everycardinal κ and every equivalence relation T on [κ]<ℵ0 with countably manyclasses, there is a c.c.c. poset P adding an injection π : κ → X which is ahomomorphism of ¬T to ¬R.

Theorem 2.5.35. Let X be a Polish space. There is a combinatorially universalBorel equivalence relation on [X]<ℵ0 with countably many classes.


Proof. Without loss of generality assume X = [ω]ℵ0 . Consider the followingrelation R on [X]<ℵ0 . Define a Borel function g : [X]<ℵ0 → [ω]<ℵ0 by g(a) ={min(x \m+ 1): x ∈ a} if a is a set of size at least two and consists of pairwisealmost disjoint sets and m is the largest number which appears in at least twoof them; g(a) = min(x) if a = {x} is a singleton; and otherwise g(a) = 0. LetR be the equivalence relation induced by the function g. We will show that Ris universal.

Let κ be a cardinal, T an equivalence relation on [κ]<ℵ0 with countably manyclasses, and let f : [κ]<ℵ0 → ω be a map inducing the equivalence relation T .Let ν : [ω]<ℵ0 → ω be a sufficiently generic map such that ν(g(0)) = f(0). LetP be the poset of all maps p such that

• dom(p) ⊂ κ is a finite set;

• for every α ∈ dom(p) the value p(α) is a nonempty subset of ω;

• for every α ∈ dom(p), ν(min(p(α))) = f(α);

• for every set a ⊂ dom(p) of size at least 2 there is a number which belongsto at least two sets p(α), p(β) for α 6= β ∈ a, and writing m for the largestsuch number, p(α) \m + 1 6= 0 holds for every α ∈ a, and ν({min(p(α) \m+ 1): α ∈ a}) = f(a).

The ordering is defined by q ≤ p if for every α ∈ dom(p), q(α) end-extends p(α),and the sets q(α) \ p(α) are pairwise disjoint for α ∈ dom(p).

Claim 2.5.36. P is c.c.c.

Proof. In fact, P is semi-Cohen in the sense of [4], but we will not need thatfact here. By the usual ∆-system arguments, it is enough to show that anytwo conditions p, q ∈ P such that p � dom(p) ∩ dom(q) = q � dom(p) ∩ dom(q),are compatible. To find the lower bound, enumerate dom(p) ∪ dom(q) as βi fori ∈ k, enumerate (dom(p) \ dom(q)) × (dom(q) \ dom(p)) as uj for j ∈ l. Usethe genericity of the function ν to build numbers m0 < m1 < · · · < ml−1 andpairwise distinct numbers nji for i ∈ k and j ∈ l so that

• m0 > max(⋃

rng(p) ∪⋃

rng(q));

• mj < nji < mj−1 for every j ∈ l;

• for every set a ⊂ dom(p) ∪ dom(q), ν({nji : βi ∈ a}) = f(a).

The lower bound is then a function r defined by dom(r) = dom(p)∪dom(q), forα ∈ dom(p), α = βi set r(α) = p(α) ∪ {nji : j ∈ l} ∪ {mj : α appears in the pair

uj}. Similarly, for α ∈ dom(q), α = βi set r(α) = q(α) ∪ {nji : j ∈ l} ∪ {mj : αappears in the pair uj}. It is not difficult to check that r ≤ p, q as required.

Claim 2.5.37. The set Dα = {p ∈ P : α ∈ dom(p)} is dense in P for everyα ∈ κ.


Proof. Let α ∈ κ and p ∈ P ; we must produce q ≤ p such that α ∈ dom(q).Enumerate dom(p) as βi for i ∈ k and write α = βk. Use the genericity of thefunction ν to find numbers m < m0 < m1 < . . .mk−1 and pairwise distinctnumbers nji for i ∈ k, j ∈ k + 1 so that

• ν(m) = f(α) and m0 > max⋃

rng(p);

• mi < nji for every j ∈ k + 1;

• for every nonempty set a ⊂ dom(p) ∪ {α} and every i ∈ k, ν({nji : βj ∈a}) = f(a).

Once this is done, just consider the function q defined by dom(q) = dom(p)∪{α}, s(α) = {m,mi : i ∈ k, nki : i ∈ k}, and for every i ∈ k, q(βi) = p(βi) ∪{mi, n

ij : j ∈ k}. It is not difficult to observe that q ∈ P and q ≤ p as desired.

Now, if G ⊂ P is a generic filter, then in V [G] let π : κ → P(ω) be definedby π(α) =

⋃p∈G p(α). The claims show that f = ν ◦ g ◦ π, in particular π is a

homomorphism of ¬T to ¬R.

Example 2.5.38. Let X be a Polish space and R a combinatorially universalequivalence relation on [X]<ℵ0 with countably many classes. Let n > 0 bea number. Let Gn be the Borel hypergraph on X consisting of all finite setsa ⊂ X which can be written in more than 2n−1 ways as a union of two distinctR-equivalent sets. Then

1. κ(EGn) ≤ ℵn+1;

2. if Martin’s Axiom for ℵn holds, then the equality is attained.

Proof. Fix the number n. The computations depend on and are motivatedby the following results of Komjath and Shelah [60]. Let κ = ℵn. For everyequivalence relation T on [κ]<ℵ0 with countably many classes, there is a finiteset a ⊂ κ which can be written in at least 2n−1 ways as a union of two distinctT -related sets. In addition, if Martin’s Axiom for κ holds then there is anequivalence relation T on [κ]<ℵ0 with countably many classes such that everyfinite set a ⊂ κ can be written in at most 2n− 1 ways as a union of two distinctT -related sets.

For (1), the first part of this result shows that there is no Gn-anticlique ofsize κ+. For (2), use Martin’s Axiom and the second part of the Komjath–Shelah result to find the equivalence relation T on [κ]<ℵ0 with countably manyclasses as above, and use the universality of the relation R to find an injectivehomomorphism π : κ → X of ¬T to ¬R. It is immediate that rng(π) ⊂ X isa Gn-anticlique of size κ. Proposition 2.5.27 then implies that κ(EGn) ≥ κ+ =ℵn+1.

Much more complicated effects can be realized if the hypergraph G is allowedto have infinite edges. We conclude this section with an example of this type.


Definition 2.5.39. Let X be a Polish space. A Borel equivalence relation R onX2 with countably many classes is combinatorially universal if for every ordinalκ and every equivalence relation T on κ2 with countably many classes, there is ac.c.c. poset P adding an injective function π : κ→ X which is a homomorphismof ¬T to ¬R.

Theorem 2.5.40. There is a combinatorially universal Borel equivalence rela-tion with countably many classes on X2 for every Polish space X.

Proof. Without loss of generality, assume X = P(ω). Let ω =⋃n,m∈ω an,m

be a partition of ω into infinite sets. For almost disjoint sets b, c ⊂ ω suchthat b is lexicographically less than c define f(b, c) = n and f(c, b) = m ifmax(b ∩ c) ∈ an,m, in other cases define f(b, c) = 0. Let R be the equivalencerelation induced by the function f . We will show that R is combinatoriallyuniversal.

Fix a cardinal κ and a function g : κ2 → ω which induces an equivalencerelation T with countably many classes. Define the poset P as the collection ofall functions p such that

• dom(p) ⊂ κ is a finite set;

• rng(p) consists of finite subsets of ω such that neither of them is an initialsegment of another;

• for every α 6= β such that p(α) is lexicographically smaller than p(β), theset p(α) ∩ p(β) is nonempty, and its maximum belongs to the set am,nwhere g(α, β) = m and g(β, α) = n.

The ordering on P is defined by q ≤ p if dom(p) ⊂ dom(q), for every α ∈ dom(p)the set p(α) is an initial segment of q(α), and the sets {q(α)\p(α) : α ∈ dom(p)}are pairwise disjoint. The following routine claims complete the proof of thetheorem.

Claim 2.5.41. The poset P is c.c.c.

Proof. By the usual ∆-system arguments it is only necessary to show that anytwo conditions p, q ∈ P such that p � dom(p)∩dom(q) = q � dom(p)∩dom(q) arecompatible in the poset P . Strengthening the conditions p, q on dom(p)\dom(q)and dom(p) \ dom(q) respectively if necessary, we may assume that no set inrng(p)∪ rng(q) is an initial segment of another. Enumerate (dom(p)\dom(q))×(dom(q)\dom(p)) as ui for i ∈ j and find pairwise distinct numbers mi for i ∈ jsuch that

• if ui = 〈α, β〉 and p(α) is lexicographically smaller than q(β) then mi ∈am,n where g(α, β) = m and g(β, α) = n;

• if ui = 〈α, β〉 and p(α) is lexicographically greater than q(β) then mi ∈am,n where g(α, β) = n and g(β, α) = m;

• all numbers mi are greater than max(⋃

rng(p) ∪⋃

rng(q)).


In the end, let r be the function defined by dom(r) = dom(p) ∪ dom(q), for allα ∈ dom(p) let r(α) = p(α) ∪ {mi : α appears in ui}, and for all β ∈ dom(q)let r(β) = q(β) ∪ {mi : β appears in ui}. It is not difficult to check that r is acommon lower bound of the conditions p, q as desired.

Claim 2.5.42. For every α ∈ κ the set Dα = {p ∈ P : α ∈ dom(r)} is dense inP .

Proof. Let α ∈ κ be an ordinal and p ∈ P be a condition; we must producea condition q ≤ p such that α ∈ dom(q). It will be the case that q(α) ∩max(

⋃rng(p)) + 1 = 0; this way, q(α) will be lexicographically smaller than all

q(β) for β ∈ dom(p). List dom(p) as βi for i ∈ j, and find pairwise distinctnumbers mi for i ∈ j so that

• mi ∈ am,n where g(α, β) = m and g(α, β) = n;

• each mi is greater than max(⋃

rng(p)).

Then, let q be the function defined by dom(q) = dom(p) ∪ {α} and q(βi) =p(βi) ∪ {mi} and q(α) = {mi : i ∈ j}. It is immediate that the condition qworks.

Now it is easy to see that if G ⊂ P is a generic filter, the function π : κ →P(ω) defined by π(α) =

⋃{p(α) : p ∈ G} induces a homomorphism of ¬T to

¬R as desired.

Example 2.5.43. Let X be a Polish space and let R be a combinatoriallyuniversal Borel equivalence relation on X2 with countably many classes. Let Gbe the hypergraph on X consisting of all unions b0 ∪ b1 where b0, b1 are infinitesets such that b0 × b1 is a subset of a single R-class.

1. if Chang’s conjecture holds, then κ(EG) ≤ ℵ2;

2. if Martin’s Axiom for ℵ2 holds then Chang’s conjecture is equivalent toκ(EG) ≤ ℵ2.

Proof. The argument is based on and motivated by two results of Todorcevic[92]. Namely, if Chang’s conjecture holds, then for every partition of ω2

2 intocountably many pieces, one piece of the partition contains a product of infinitesets. In addition, if Martin’s Axiom for ℵ2 holds and Chang’s conjecture fails,then there is a partition of ω2

2 into countably many pieces such that no piece ofthe partition contains a product of infinite sets.

Now, for (1) use the first result of Todorcevic to argue that under Chang’sconjecture there is no G-anticlique of size ℵ2, so by Proposition 2.5.27, κ(EG) ≤ℵ2. For (2), suppose that Martin’s Axiom holds and Chang’s conjecture fails.By the second result of Todorcevic, find an equivalence relation T on ω2

2 withcountably many classes without any monochromatic infinite rectangles. Use theuniversality of the equivalence relation R to find an injection π : ω2 → X whichis a homomorphism of ¬T to ¬R. Observe that rng(π) is a G-anticlique of sizeℵ2. By Proposition 2.5.27, κ(EG) > ℵ2.

2.6. RESTRICTIONS ON PARTIAL ORDERS 65

2.6 Restrictions on partial orders

Given an analytic equivalence relation E on a Polish space X, it may be informa-tive to investigate which posets can carry nontrivial E-pinned names. After all,essentially all pinned names discussed in this chapter naturally live on collapseposets, so one may easily (and wrongly) assume that no forcing sophisticationis needed when it comes to the investigation of the virtual realm. This sectioncontains several theorems on this topic.

First of all, there are some partial orders which can never carry a nontrivialpinned name.

Definition 2.6.1. [29] A poset P is reasonable if for every ordinal λ and forevery function f : λ<ω → λ in the P -extension there is a set a ⊂ λ which isclosed under f , belongs to the ground model, and is countable in the groundmodel.

In particular, all c.c.c. and all proper forcings are reasonable. Good examples ofunreasonable forcings are posets which collapse ℵ1, Namba forcing and Prikryforcing.

Theorem 2.6.2. Let E be an analytic equivalence relation on a Polish spaceX. If P is a reasonable forcing and τ is an E-pinned name on P , then τ istrivial.

Proof. Suppose that P is a reasonable poset and τ is an E-pinned name on P .We will produce a condition p ∈ P and a point x ∈ X such that p τ E x.Towards this end, choose a large structure and use the reasonability of P to finda countable elementary submodel M of it containing P,E and τ and a conditionp ∈ P such that p G∩M is generic over M , where G is the canonical P -namefor its generic ultrafilter. As M is countable, there is a filter H ⊂ P ∩M genericover M in the ground model V . Let x = τ/H ∈ X. Proposition 2.1.5 applied tothe model M and the filters H and G∩M now says that p x E τ , completingthe proof.

The key feature of partial orders from the point of view of existence of pinnednames is collapsing ℵ1, as the following theorem shows:

Theorem 2.6.3. Let E be an analytic equivalence relation on a Polish spaceX. Exactly one of the following occurs:

1. E is pinned;

2. for every poset P collapsing ℵ1, P carries a nontrivial E-pinned name.

Proof. Clearly (1) implies the negation of (2). For the difficult direction, supposethat (1) fails and work to confirm (2). Let τ be a nontrivial E-pinned nameon some poset P . Let 〈Mα : α ∈ ω1〉 be a continuous ∈-tower of countableelementary submodels of a large structure containing X,E, τ, and P . Let M =⋃αMα, let Q = P ∩M and let σ = τ ∩M .


First, observe that the pair 〈Q, σ〉 is an E-pin: by elementarity, M |= Q×Q σleft E σright, and by the Mostowski absoluteness between the generic extensionsof M and V , V |= Q × Q σleft E σright as well. We will now prove that thepair 〈Q, τ〉 is a non-trivial E-pin.

Assume towards a contradiction that there exists a point x ∈ X such thatQ σ E x. We will show that there then must be y ∈M ∩X which is E-relatedto x. Then Q σ E y, by the Mostowski absoluteness between the Q-extensionsof M and V M |= P τ E y, and this will contradict the elementarity of themodel M and the nontriviality of the name τ .

To find the point y ∈ M ∩ X, let N be a countable elementary submodelof a large structure containing 〈Mα : α ∈ ω1〉, Q, x. Since the tower of models〈Mα : α ∈ ω1〉 is continuous, there is a limit ordinal α ∈ ω1 such that Mα =N∩M . Let Qα = Q∩Mα = P∩Mα and σα = σ∩Mα = τ∩Mα. By elementarityof the model N and analytic absoluteness between the Qα-extension of N andV , Qα σα E x. Since Qα = P ∩Mα and σα = τ ∩Mα, both Qα, τα belongto the model Mα+1. By the elementarity of the model Mα+1, there must be apoint y ∈ X ∩Mα+1 such that Qα σα E y (since the point x is such). By thetransitivity of E, it follows that x E y. The point y ∈Mα+1 ⊂M works.

Now, suppose that R is a poset collapsing ℵ1. Since |M | = ℵ1, in the R-extension there is a filter Q-generic over M . Let H be an R-name for such afilter and let ν be the R-name for σ/H. By the Mostowski absoluteness betweenthe generic extensions of M and V again, it follows that Q × R σ E ν. Wehave just proved that ν is a nontrivial E-pinned name on R.

Theorem 2.6.3 does not rule out the possibility that some ℵ1 preservingposets carry nontrivial E-pinned names. This does not occur for orbit equiv-alence relations. In them, every pinned name is inescapably connected with acollapse of a certain cardinal to ℵ0. To state this in the most informative way,we introduce a piece of notation. Let E be an analytic equivalence relation ona Polish space X, and let c be a virtual E-class. We write κ(c) for the smallestcardinality of a poset P such that there is an E-pinned P -name σ such that theE-pin 〈P, σ〉 belongs to c.

Theorem 2.6.4. Let E be an equivalence relation on a Polish space X, Borelreducible to an orbit equivalence relation of a Polish group action. Let c be avirtual E-class and write κ = κ(c). The following are equivalent for every posetP :

1. there is an E-pinned P -name σ such that the E-pin 〈P, σ〉 belongs to c;

2. P |κ| = ℵ0.

Proof. (2) implies (1) is the easier direction, and it does not use the assumptionon the equivalence relation E. Suppose that P |κ| = ℵ0|. Let 〈R, τ〉 bean E-pin in c such that |R| = κ. Let M be an elementary submodel of alarge structure such that |M | = κ, R ⊂ κ, and R, τ ∈ M . By the collapsingassumption, there is a P -name η for a filter on R which is generic over the model

2.6. RESTRICTIONS ON PARTIAL ORDERS 67

M . Let σ be a P -name for τ/η. We claim that σ is E-pinned and moreover〈P, σ〉 ∈ c.

To this end, let G ⊂ P and H ⊂ R be mutually generic filters; we must showthat τ/G E σ/H. To see this, observe that the filters η/G ⊂ R and H ⊂ Rare mutually generic over the model M , so M [η/G,H] |= σ/η/G E σ/H as thename σ is E-pinned and M is an elementary submodel. σ/η/G = τ/G E σ/Hthen follows by the Mostowski absoluteness between the models M [η/G,H] andV [G,H].

For the harder direction (1) implies (2), first fix a Polish group Γ, an actionof Γ on a Polish space Y , and a Borel function h : X → Y which is a reduction ofE to the orbit equivalence relation F induced by the action of Γ. Suppose thatσ is an E-pinned name on P such that the E-pin 〈P, σ〉 belongs to c. Considerthe Cohen poset PΓ of nonempty open subsets of Γ, with its associated nameγgen for a generic element of Γ. Consider the product PΓ × P and the posetQ, the subset of the complete Boolean algebra of PΓ ×P which is generated bythe name τ for γgen · h(σ), an element of the space Y . Since the poset PΓ iscountable, it preserves all cardinals. Therefore, to show that P collapses κ, itwill be enough to show that Q collapses κ.

Claim 2.6.5. Let H ⊂ Q be a generic filter. In V [H], PΓ forces the pointγgen · τ/H ∈ Y to be Q-generic over V [H].

The difficulty resides in the assertion that the point is forced to be Q-genericover V [H] and not just over V .

Proof. Let γ0, γ1, γ2 ∈ Γ and K0,K1 ⊂ P be PΓ-generic points and genericfilters on P respectively, which are in addition mutually generic. Since σ isan E-pinned name, σ/K0 E σ/K1 holds. Since h is a reduction of E to F ,h(σ/K0) F h(σ/K1) holds. Writing y0 = h(σ/K0) and y1 = h(σ/K1), theremust be a group element δ ∈ Γ such that δγ0 · y0 = γ1 · y1 in the modelV [γ0,K0][γ1,K1]. Observe that both points γ2δ and γ2γ1 are PΓ-generic overthe model V [γ0,K0][γ1,K1] since γ2 is, and multiplication by δ (or γ1) from theright induces an automorphism of the poset PΓ.

It follows that γ2δ is PΓ-generic, and the pair γ2γ1,K1 is PΓ × P -generic,both over the model V [γ0,K0]. Observe that γ2δγ0 · y0 = γ2γ1 · y1 by the initialchoice of δ. The claim now follows by considering the filter H ⊂ Q (generic overV ) obtained from the point γ0 · y0 ∈ Y , and the point γ2δ ∈ Γ (PΓ-generic overV [H]), and the forcing theorem.

Now, let H ⊂ Q be a generic filter, and work in V [H]. Since the posetPΓ has a countable dense subset and it introduces a generic for Q, it must bethe case that Q has a countable dense subset as well. Back in V , the forcingtheorem says that Q forces Q to have a countable dense subset. Fix a Q-name η for a function from ω to Q whose range is forced to be dense, and letD = {q ∈ Q : ∃r ∃m ∈ ω r η(m) = q}. It will be enough to show that |D| ≥ κand Q |D| = ℵ0.


The latter statement is easier: Q forces D to be the range of the partialfunction f on ω × ω defined by f(n,m) = q if η(n) η(m) = q. For the formerstatement, first note that D ⊂ Q is dense since D is forced to contain the rangeof η. It follows that D and Q viewed as posets give the same extensions and τ isreally a D-name. By the Mostowski absoluteness between the PΓ×P -extensionand the D-extension, there must be a D-name χ for an element of X suchthat D h(χ) F τ . Since h is a reduction, it follows that 〈Q,χ〉 is an E-pinequivalent to 〈P, σ〉. By the initial choice of the cardinal κ, |Q| ≥ κ follows.

Corollary 2.6.6. Let E be an analytic equivalence relation on a Polish spaceX, Borel reducible to an orbit equivalence relation. If E is not pinned then thefollowing are equivalent for any partial order P :

1. P carries a nontrivial E-pinned name;

2. P collapses ℵ1 to ℵ0.

Proof. (2) implies (1) by Theorem 2.6.3. For the opposite implication, let τbe a nontrivial E-pinned name on P . Let κ be the cardinal of Theorem 2.6.4associated with the virtual E-class containing 〈P, τ〉. Note that κ cannot beℵ0 since then the Cohen poset would carry an E-pinned name equivalent to τ ;however, all E-pinned names on the Cohen poset are trivial by Theorem 2.6.2.Note also that P collapses κ to ℵ0 since it realizes the virtual E-class associatedwith 〈P, τ〉. In conclusion, P collapses some uncountable cardinal to ℵ0, inparticular it must collapse ℵ1.

Example 2.6.7. Consider the equivalence relation F2 on (2ω)ω. This is clearlyan orbit equivalence relation. Every E-pinned name is represented by a setA ⊂ 2ω by Example 2.3.5. The cardinal κ of Theorem 2.6.4 is equal to |A|.

Example 2.6.8. Let E be the equivalence relation on X = (P(ω))ω definedby x0 E x1 if rng(x0) and rng(x1) generate the same filter on ω. Suppose thatin V , there is a modulo finite strictly decreasing sequence a = 〈aα : α ∈ ω2〉 ofsubsets of ω. Let P be the Namba forcing. It is well-known that P preserves ℵ1

and adds a cofinal sequence σ : ω → ωV2 to ω2. Let τ be the P -name for a◦σ. Itis immediate that τ is a nontrivial E-pinned name on P . Thus, the assumptionthat E be reducible to an orbit equivalence in Corollary 2.6.6 cannot be omitted.

2.7 Absoluteness

This section compiles the available information regarding the absoluteness ofvarious notions regarding the virtual equivalence classes. The most substantialand useful statement is Corollary 2.7.3: if E is a Borel equivalence relation thenits pinned status is absolute among all generic extensions.

Theorem 2.7.1. Suppose that E is a Borel equivalence relation on a Polishspace X. The following are equivalent:

2.7. ABSOLUTENESS 69

1. E is pinned;

2. For every ω-model M of ZFC containing the code for E, M |= E is pinned.

Proof. For simplicity assume X = ωω. The implication (2)→(1) is trivial. If(1) fails, then E is not pinned, and the failure of (2) is witnessed by M = V .

The implication (1)→(2) is more difficult. We start with an abstract claimwhich can be used in several other absoluteness results, and which appears inseveral less efficient versions in existing literature. Say that an ω-model Nof ZFC is correct about stationary sets if for all sets a, b ∈ N , if N |= a isuncountable then in fact a is uncountable, and if N |= b ⊂ [a]ℵ0 is a stationarysystem of countable sets, then in fact b is a stationary system of countable sets.Note that as N is an ω-model, for any set c, if N |= c is countable then in factc is countable.

Claim 2.7.2. [5, Theorem 1.12], [51, Section 4] Let M be a countable ω-modelof a large fragment of ZFC. Then there is an elementary embedding j : M → Nsuch that N is an ω-model correct about stationary sets.

Proof. We start with setting up a bookkeeping tool. By a classical result ofSolovay, ω1 can be decomposed into ℵ1 many stationary sets. Thus, there isa surjective function f : ω1 → ω1 × ω such that preimages of singletons arestationary.

Now, by recursion on α ∈ ω1 build countable ω-models Mα of ZFC andmaps jβα : Mβ → Mα for all β ∈ α. With each model Mα we also pick asurjection πα : ω →Mα arbitrarily. The following are the recursive demands onthe construction:

• M = M0 and 〈jγβ : Mγ → Mβ : γ ∈ β ∈ α〉 is a commuting system ofelementary embeddings;

• if α ∈ ω1 is limit then Mα is the direct limit of the previous models andembeddings;

• whenever α ∈ ω1 is an ordinal such that f(α) = 〈β, n〉 and β ∈ α andMβ |= b = πβ(n) is a stationary system of countable sets on a =

⋃b, then

writing c = j′′αα+1jβα(a) we get that Mα+1 contains c and Mα+1 |= c ∈jβα+1(b).

It is only necessary to show how the last item is arranged. Thus, suppose thatits assumptions are satisfied at α ∈ ω1. Work in the model Mα and considerthe poset P of all stationary subsets of b, ordered by inclusion. Since the modelMα is countable, there is a filter G ⊂ P generic over the model Mα. Letjαα+1 : Mα → Mα+1 be its associated generic ultrapower as in [45, Lemmas22.13 and 14]. Then Mα+1 will be again an ω-model. Note that the set cbelongs to the model Mα+1 as it is represented by the identity function; it alsohas the requested properties by the Los theorem.

In the end, let j : M → N be the direct limit of the system of elementaryembeddings produced during the recursion; we claim that this embedding works


as required. Suppose then that M |= b is a stationary system of countable setson a =

⋃b. To show that b is indeed stationary, let d be a closed unbounded

set of countable subsets of a, and work to show that d ∩ b 6= 0 holds. Fix anordinal β ∈ ω1 and a number n ∈ ω such that b = jβω1

(πβ(n)). Let K be acountable elementary submodel of a large substructure, containing in particularthe set d, the ordinal β, and the system of elementary embeddings constructedabove, and such that f(α) = 〈β, n〉 where α = K ∩ ω1. Such a model exists bythe initial choice of the function f . Clearly K ∩ a ∈ d; we will be done once weshow that K ∩ a ∈ b.

To see this, let c = j′′αα+1jβα(⋃πβ(n)). By the elementarity of the em-

bedding jα+1ω1 and the last item above, it is the case that jα+1ω1(c) ∈ b.Also, jα+1ω1

(c) = j′′α+1ω1c = j′′αω1

jβα(⋃πβ(n)), By the elementarity of the

model K and the second item above, the latter expression is exactly equal toK ∩ jβω1

(⋃πβ(n)) = K ∩ a. This completes the proof.

Now, back to the theorem. Suppose that (2) fails. Fix a ω-model M of ZFCcontaining the code for E such that M |= E is unpinned; taking an elementarysubmodel if necessary we may assume that M is countable. Let j : M → N be anelementary embedding from the claim. By elementarity, N |= E is unpinned,so there is a poset P and a P -name τ such that N |= τ is a nice nontrivialE-pinned name. We will show that in fact τ really is a nontrivial E-pinnedname.

First of all, τ indeed is E-pinned. If G0, G1 ⊂ P are mutually generic filtersthen N [G0, G1] |= τ/G0 E τ/G1 holds, and by the Borel absoluteness betweenN [G0, G1] and V [G0, G1], τ/G0 E τ/G1 holds in V [G0, G1] as required. Toshow that τ is a nontrivial E-pinned name, assume towards a contradictionthat there is a point x ∈ X such that P τ E x; the difficulty resides in thepossibility that x /∈ N .

Let κ be some large cardinal in the model N . Let K be a countable elemen-tary submodel of a large structure containing in particular N,P, τ , and κ, suchthat c = K ∩ N ∩ Vκ belongs to the model N and is countable there. Such amodel K exists since the model N is correct about the stationarity of the setit views as [Vκ]ℵ0 . Work in the model N . Since c is an elementary submodelof Vκ, it follows that σ = τ ∩ c is an E-pinned name on the poset Q = P ∩ c.Since the poset Q is countable, by Theorem 2.6.2 below applied in the modelN , there must be a point y ∈ N such that Q σ E y. Now, work in the modelK and observe that Q σ E x. It follows that x E y. Now, back in the modelN it must be the case that P τ E y, contradicting the assumption that τ isa nontrivial E-pinned name.

Corollary 2.7.3. Let E be a Borel equivalence relation on a Polish space X.The statement “E is pinned” is absolute between all generic extensions.

Proof. The validity of item (2) of Theorem 2.7.1 does not change if one only con-siders countable models. This follows from an immediate downward Lowenheim-Skolem argument. The countable model version of (2) is a coanalytic statement,


and as such it is absolute among all forcing extensions by the Mostowski abso-luteness.

The absoluteness of the pinned status of analytic equivalence relations doesnot allow a similar sweeping statement. Consider a Π1

1 set A ⊂ 2ω which con-tains no perfect subset, but is uncountable in the constructible universe L [45,Corollary 25.37]. Let E be the analytic equivalence relation on X = (2ω)ω de-fined by x0 E x1 if either both rng(x0), rng(x1) have nonempty intersection withthe complement of A, or rng(x0) = rng(x1). In L, the equivalence relation Eis not pinned, as witnessed by the Coll(ω,A)-name for the generic enumerationof A. At the same time, in the Coll(ω, 2ω)-extension of L, the (reinterpreta-tion of the) set A is countable, containing only constructible elements, and inconsequence the equivalence relation E is smooth, so pinned. On the otherhand, in the presence of sufficiently large cardinals the pinned status of everyanalytic equivalence relation is absolute by a proof similar to the proof of The-orem 2.7.1 where we adjust Claim 2.7.2 to input a well-founded model M ofZFC plus the statement that there is a propoer class of Woodin cardinals, andoutput a well-founded model N correct about stationary sets. The adjustmentneeds the classical well-foundedness results regarding the well-foundedness ofthe stationary tower ultrapower in [99, 64]. We refrain from providing furtherdetail.

Another natural question concerns the computation of the various cardinalinvariants of equivalence relations in inner models. In particular, we would liketo see that if M ⊂ N are transitive models of ZFC containing the code foran analytic equivalence relation E, then (κ(E))M ≤ (κ(E))N . Interestinglyenough, this is not clear, and we only have a positive answer in the case of orbitequivalence relations and their relatives.

Theorem 2.7.4. Let E be an analytic equivalence relation on a Polish space Xalmost reducible to an orbit equivalence relation. Let M be a transitive model oflarge fragment of set theory containing the codes for E, the group action, andthe almost reduction. Then κ(E)M ≤ κ(E) holds.

Proof. To begin, let Γ be a Polish group continuously acting on a Polish spaceY , inducing an orbit equivalence relation F . Let h : X → Y be a Borel reductionof E to F . Choose Γ, Y, h so that they belong to the model M .

Suppose that the conclusion of the theorem fails. Then, writing κ = (κ(E))V ,in the model M there exists an E-pin 〈Q, τ〉 such that M |= 〈Q, τ〉 is not equiv-alent to any E-pin on a poset of cardinality less than κ. By Theorem 2.6.3applied in the model M , this means that if N is a generic extension of M , thenN |= (∃x ∈ X Q τ E x)→ |κ| = ℵ0).

Let 〈R, σ〉 be an E-pin in V such that |R| < κ and Q × R τ E σ.Let G ⊂ Q, H ⊂ R and K ⊂ PΓ be filters mutually generic over V . Writey0 = h(τ/G), y1 = h(σ/H), write γ ∈ Γ for the generic point associated withthe filter K, and let y2 = γ · y1. By the assumptions, y0, y1, and y2 ∈ Y aremutually orbit-equivalent points.


Claim 2.7.5. The point y2 ∈ Y is generic over the model M , and in the modelM [y2] there exists a point x ∈ X such that M [y2] |= Q τ E x.

Proof. There is a point δ ∈ Γ in the model V [G][H] such that y1 = δ · y0. Thepoint γ ∈ Γ is PΓ-generic over V [G][H] by the product forcing theorem. Now,multiplication by δ from the right is an automorphism of the poset PΓ, so thepoint γδ ∈ Γ is PΓ-generic over V [G][H]. Therefore, the point γδ is also PΓ-generic over the smaller model M [G], and by the product forcing theorem, Gand γδ are mutually generic objects over M . The point y2 = γδ · y0 ∈M [G, γδ]belongs to a generic extension of M , so is generic over M itself.

Now, the model M [y2] is transitive, so by Mostowski absoluteness it mustcontain a point x ∈ X such that h(x) F y2, since one such point σ/H exists inthe model V [H][K]. We claim that M [y2] |= Q τ E x. To see this, observethat the filter G ⊂ Q is generic over the model V [H][K], so also generic over thesmaller model M [y2]. Since M [G][y2] |= y0 = h(τ/G) F y2, the forcing theoremimplies that M [y2] |= Q h(τ) F y2, so τ E x.

By the second paragraph of the proof, it must be the case that M [y2] |= |κ| =ℵ0; therefore, V [H][K] |= |κ| = ℵ0. This, however, contradicts the assumptionthat in V , |R| < κ, which is equivalent to |R× PΓ| < κ.

2.8 Dichotomies

This section presents three theorems which characterize various complex fea-tures of the virtual realm in terms of dichotomies. It turns out that each ofthe dichotomies uses a preparatory proposition or two on the descriptive the-ory of forcing and model theory. These preparatory propositions are perhapsmore difficult to state properly than to prove. Nevertheless, they are unavoid-ably needed and they do not seem to be present in published literature. Wegather them in the first subsection, while the second subsection contains thedichotomies themselves.

2.8a Preliminaries

The first two auxiliary propositions concern pure model theory; they show thatthe satisfaction relation and the construction of models from indiscernibles arein a suitable sense Borel affairs. The proofs are straightforward and left to thereader. For both of them, fix a countable language and let X be the Polishspace of models of that language whose universe is ω.

Proposition 2.8.1. Suppose that f : 2ω → X is a Borel function, φ is a formulaof the language with n free variables, and gi : 2ω → ω are Borel functions forevery i ∈ n. The set {x ∈ X : f(x) |= φ(g0(x), g1(x) . . . gn−1(x))} is Borel.

Let T be a Skolemized complete consistent theory containing a linear ordering.Let S be the theory of an infinite ordered set of indiscernibles in the linear orderof T . Let Y be the space of all linear orderings on ω.

2.8. DICHOTOMIES 73

Proposition 2.8.2. There are Borel functions f : Y → X and g : Y × ω → ωsuch that for each y ∈ Y ,

1. f(y) is a model of T ;

2. the function gy : ω → ω is an order-preserving map from the ordering y tothe ordering of f(y) and its range consists of indiscernibles with theory S;

3. the model f(y) is a Skolem hull of rng(gy).

The rest of the subsection deals with models of set theory, showing thatthe construction of their generic extensions is a Borel affair. Thus, let X bethe Polish space of all models for the language with one binary relation whoseuniverse is ω. Let Y be any uncountable Polish space, serving as an index space.Let M : Y → X and P : Y → ω be Borel functions such that for each y ∈ Y ,M(y) is a model of (a large fragment of) ZF and M(y) |= P (y) is a partiallyordered set.

Proposition 2.8.3. There is a Borel function G : Y → P(ω) such that forevery y ∈ Y , G(y) is a filter on P (y) which is generic over M(y).

Proof. By induction on n ∈ ω define Borel functions fn : Y → ω so that

• for every y ∈ Y , M(y) |= f0(y) is the largest element of the poset P (y);

• for every y ∈ Y and n ∈ ω, if M(y) |= n is an open dense subset of theposet P (y), then fn+1(y) is the smallest number m such that M(y) |= mis an element of n and it is smaller that fn(y) in the poset P (y); otherwise,fn+1(y) = fn(y).

The functions defined in this way are Borel by Proposition 2.8.1. Define thefunction G : Y → P(ω) as follows: for every y ∈ ω, G(y) is the set of all msuch that M(y) |= m is an element of the poset P (y) and for some n ∈ ω,M(y) |= fn(y) is smaller than m in the poset P (y). It is clear that the functionG works.

Now, suppose that a Borel function G : Y → P(ω) as in Proposition 2.8.3 isgiven; we want to produce the generic extensions. To wit, we have to producethe generic extensions as Borel functions on Y and also the valuation functionsfor names in the ground models as Borel functions.

Proposition 2.8.4. There are Borel functions M [G] : Y → X and v : Y ×ω →ω such that for every y ∈ Y , M [G](y) is a generic extension of M(y) by G(y)and for every n ∈ ω, v(y)(n) = n/G(y) whenever M(y) |= n is a P (y)-name.

Proof. Let E be the Borel equivalence relation on Y ×ω connecting pairs 〈y0, n0〉and 〈y1, n1〉 if y0 = y1 and (denoting their common value by y) either M(y) |=neither n0, n1 is a P (y)-name or M [y] |= both n0, n1 are P (y)-names and thereis some condition p ∈ G(y) such that M(y) |= p n0 = n1. Let v : Y × ω → ω


be the Borel function defined in the following way: f(y)(n) = 0 if M(y) |= n isnot a P (y)-name, and f(y)(n) = m if M(y) |= n is a P (y)-name and, writingn′ for the smallest name E-related to n, there are exactly m-many E-classes ofP (y)-names represented by numbers smaller than n′. Let M [G] be the Borelfunction defined in the following way: M [G](y) connects the pair 〈m0,m1〉 justin case there are numbers n0, n1 ∈ ω such that M(y) |= n0, n1 are P (y)-names,f(y)(n0) = m0 + 1, f(y)(n1) = m1 + 1, and there is a condition p ∈ G(y) suchthat M(y) |= p n0 ∈ n1. The Borelness of the functions M [G] and v followsfrom Proposition 2.8.1.

The construction of the generic filters may be subject to an additional constraint.Suppose that E is an equivalence relation on a Polish space Z, coded in eachmodel M(y). Suppose that τ : Y → ω is a Borel function such that for eachy ∈ Y , M(y) |= τ(y) is a P (y)-name of an element of Z. Suppose that z : Y → Zis a Borel function. Can we find a Borel function G : Y → P(ω) such that foreach y ∈ Y , G(y) is a filter on P (y) generic over M(y), such that the valuation ofτ(y) over G(y) is E-related to z(y)? To come to a useful affirmative conclusion,we assume that the models in the range of the function M are all well-founded,and the equivalence relation E is induced as an orbit equivalence relation of acontinuous action of a Polish group Γ on the space Z. In such circumstances,we have the following:

Proposition 2.8.5. Given E,M,P, τ as above:

1. the set B = {y ∈ Y : there is a filter G ⊂ P (y) generic over the modelM(y) such that τ(y)/G is E-related to z(y)} is Borel;

2. there is a Borel function G : B → P(ω) such that for every y ∈ B, G(y) ⊂P (y) is a filter generic over M(y) such that τ(y)/G(y) is E-related toz(y).

Proof. Let PΓ be the poset of all nonempty open subsets of Γ ordered by in-clusion, with its name γ for a generic element of Γ. Use Proposition 2.8.1 tofind Borel functions Q, σ : Y → ω so that for every y ∈ Y , M(y) |= Q(y) is thecomplete subalgebra of the complete Boolean algebra of P (y) × PΓ generatedby the name σ(y) = γ · τ(y).

Let D = {〈y, δ,G〉 ∈ Y ×Γ×P(ω) : G ⊂ Q(y) is a filter generic over the modelM(y) such that ν(y)/G = δ · z(y)}. This is a Borel set by Proposition 2.8.1.The projection of D into the Y coordinate is the set B.

Claim 2.8.6. For each pair 〈y, δ〉 ∈ Y × Γ, the vertical section Dyδ is eitherempty or a singleton.

Proof. This uses the wellfoundedness of the model M(y). Suppose that the sec-tion Dyδ is nonempty, containing some filter G. As M(y) |=“Q(y) is completelygenerated by the name σ(y)”, the filter G can be recovered by transfinite induc-tion using infinitary Boolean expressions in M applied to δ · z(y); therefore, itis unique.

2.8. DICHOTOMIES 75

Write C ⊂ Y × Γ for the projection of D into the first two coordinates. Sinceone-to-one projections of Borel sets are Borel [55, Theorem 15.1], the set C isBorel.

Claim 2.8.7. For each point y ∈ Y , the section Cy either empty or else comea-ger in Γ.

Proof. To simplify the notation, fix y ∈ Y and omit the argument y from theexpressions like M(y), σ(y) . . . . Suppose that the Γ-section Cy is nonempty.Thus, there is a filter G ⊂ P be a filter generic over M such that τ/G isE-related to z. Let A ⊂ Γ be the set of all elements of the group Γ whichare PΓ-generic over the model M [G]; this is a co-meager set. Let δ ∈ Γ bean element of the group such that τ/G = δ · z. Then the vertical section Cycontains the set A · δ which is co-meager as the right multiplication by δ is aself-homeomorphism of the group Γ.

As the category quantifier yields Borel sets [55, Theorem 16.1], the set B as theprojection of C into the first coordinate is Borel. Borel sets with nonmeagervertical sections allow Borel uniformizations [55, Theorem 18.6], so there isa Borel uniformization f : B → Γ of C. As the set D has singleton verticalsections, it is itself its uniformization g : C → P(ω). Let H : B → P(ω) be thefunction defined by H(y) = g(y, f(y)). Thus, for every y ∈ B, H(y) ⊂ Q(y) isa filter generic over M(y) such that σ(y)/H(y) is E-related to z(y).

Now, let R : Y → ω be a Borel function such that for every y ∈ 2ω, M(y) |=R(y) is a name for the quotient poset P (y)× PΓ/Q(y). Use Propositions 2.8.3and 2.8.4 to find a Borel function K : B → P(ω) such that K(y) ⊂ R(y) is afilter generic over the model M(y)[H(y)]. Let G : B → P(ω) be a Borel functionindicating a filter on P (y) which is the P (y)-coordinate the composition ofH(y) ∗K(y). The function G has the required properties.

Corollary 2.8.8. Suppose that Γ is a Polish group continuously acting on aPolish space X, inducing an orbit equivalence relation E. Let M be a countabletransitive model of set theory containing a code for the action, let P ∈ M be aposet and τ ∈M be a P -name for an element of the space X. Then

1. the set B = {x ∈ X : ∃G ⊂ P generic over M such that τ/G = x} isBorel;

2. the equivalence relation E � B is Borel.

Proof. (1) is immediate from the proposition. To see (2), consider the set C ={〈x, y〉 ∈ X2 : ∃G × H ⊂ P × PΓ generic over M such that x = τ/G andy = (γgen/H) · x}. The set C is Borel by the proposition again. Now, forx, z ∈ B, x E z just in case the set Axz = {γ ∈ Γ: 〈x, γ · z〉 ∈ C} is comeager inΓ. To see this, note that if x E z fails then the set Axz ⊂ Γ is actually empty,and if x E z holds with δ · x = z, then the Axz ⊃ {γ ∈ Γ: γ · δ−1 is PΓ-genericover M [x]} and the latter set is comeager since it is a shift of the comeager setof all points in Γ which are PΓ-generic over the model M [x].


Finally, observe that the equivalence relation E � B is obtained by an ap-plication of the category quantifier to the Borel set {〈x, z, γ〉 ∈ B ×B × Γ: γ ∈Axz〉}, so is Borel by [55, Theorem 16.1].

2.8b Results

The first dichotomy characterizes the existence of the pinned cardinal in analyticequivalence relations.

Theorem 2.8.9. Assume that there is a measurable cardinal. Let E be ananalytic equivalence relation on a Polish space X. The following are equivalent:

1. κ(E) =∞;

2. Eω1≤a E.

Proof. (2) implies (1) by Example 2.4.6 and Theorem 2.5.4. The large cardi-nal assumption is not needed for this direction. For the (1)→(2) implication,suppose that κ(E) =∞. Let κ be a measurable cardinal.

Claim 2.8.10. There exist a poset P of size κ and an E-pinned P -name τ suchthat τ is not E-related to any name on a poset of size less than κ.

Proof. Since κ(E) = ∞, there exist a poset Q and an E-pinned Q-name σwhich is not E-equivalent to any name on a poset of size less than κ. Choosean elementary submodel M of a large structure such that |M | = κ, Vκ ⊂ Mand Q, σ ∈ M . Let P = Q ∩M and let τ = σ ∩M . We claim that P, τ worksas required.

Indeed, if G ×H ⊂ P × P is a filter generic over V , then it is also genericover M , by the elementarity of M and the forcing theorem in M M [G,H] |=τ/G E τ/H, and by the Mostowski absoluteness between M [G,H] and V [G,H],V [G,H] |= τ/G E τ/H. This proves that the name τ is E-pinned. If R is aposet in V of size < κ and ν is an E-pinned R-name and G×H ⊂ P × R is ageneric filter over V , then R, ν ∈ M , the filter G ×H also generic over M , bythe elementarity of M and the forcing theorem in M M [G,H] |= ¬τ/G E ν/H,and by the Mostowski absoluteness between M [G,H] and V [G,H], V [G,H] |=¬τ/G E σ/H. This proves that ν E τ fails and completes the proof of theclaim.

Choose a poset P on κ and an E-pinned name τ as in the claim. Choose alarge enough cardinal θ and skolemize the structure 〈Hθ,∈, P, τ〉 of sets whosetransitive closures have cardinality less than θ. Let T be the theory of theskolemized structure. Let j : V = M0 → M1 be an ultrapower embeddingassociated with some normal measure on the measurable cardinal κ, and letjn : V →Mn be its iterations of length n for n ≤ ω. The ordinals {jn(κ) : n ∈ ω}form a sequence of indiscernibles in the model jω(Hθ). Let S be the theory ofthe indiscernibles in this structure. Let Y be the space of all linear orderingson ω, and let Z be the space of all binary relations on ω. By Propositions 2.8.2

2.8. DICHOTOMIES 77

and 2.8.3, there are Borel functions M : Y → Z, G : Y ×M → P(ω), P : Y → ωand τ : Y → ω such that whenever y ∈ Y is a linear ordering then M(y) is amodel of T which is a Skolem hull of indiscernibles of ordertype y satisfying thetheory S, P (y) is its version of the poset P , and τ(y) is its version of the name τ ,and G(y) ⊂ P (y) is a filter generic over the model M(y). Note that all modelsM(y) are ω-models, so compute the Polish space X correctly. Let k : Y → X bethe function defined by k(y) = τ(y)/G(y), which is Borel by Proposition 2.8.4.It will be enough to show that k is a Borel reduction of Eω1 to E on the set ofy ∈ Y which are well-orders.

Suppose first that y0, y1 ∈ Y are well-orders of the same length. Thenthe models M(y0),M(y1) are wellfounded and isomorphic. Write N for theircommon transitive isomorph and Q, σ for its version of the poset P and thename τ . Thus, N |= σ is an E-pinned Q-name. The filters G(y0), G(y1) ⊂ Qare separately generic over N . Let G ⊂ Q be a filter generic over both countablemodels N [G(y0)] and N [G(y1)]. By the product forcing theorem, the filtersG(y0)×G and G(y1)×G are both Q×Q-generic over N . As N |= σ is an E-pinned name, it follows that N [G(y0), G] |= σ/G(y0) E σ/G and N [G(y1), G] |=σ/G(y1) E σ/G. By the Mostowski absoluteness between these two models andV , and the transitivity of the relation E, σ/G(y0) = k(y0) E k((y1) = σ/G(y1)follows.

Suppose now that y0, y1 ∈ Y are well-orders of different lengths; say that y0

is shorter than y1. Let N0 be the transitive isomorph of M(y0), let κ0, P0, σ0

be the versions of κ, P , and τ in the model N0, and let G0 ⊂ P0 be the filtergeneric overN0 indicated byG(y0). Similar notation prevails for subscript 1. Leti : N0 → N1 be the elementary embedding obtained by sending the indiscerniblesof N0 to an initial segment of the indiscernibles of N1. Note that the criticalpoint of i is exactly κ0, so P0 ∈ N1 as P0 = P1 � κ0. Also, N1 |= |P0| < κ1. Finda filter G ⊂ P0 generic over both the countable models N0[G0] and N1[G1]. SinceN0 |= σ0 is E-pinned, the Mostowski absoluteness between V and N0[G,G0]implies that σ0/G0 E σ0/G holds. Since N1 |= σ1 is not E-equivalent to anyname on a poset of size < κ1, in particular to σ0, it follows from the Mostowskiabsoluteness between V and N1[G,G1] that σ0/G E σ1/G1 fails. In conclusion,k(y0) = σ0/G0 E σ1/G1 = k(y1) fails as required.

The status of a Borel equivalence relation as pinned/unpinned may be absolutebetween models of ZFC by Corollary 2.7.3. However, if one dares to look atchoiceless models, a much more colorful picture comes to sight. The simplestdescription of the pinned status occurs in the Solovay model, where it actuallyobeys a former conjecture of Kechris [48, Question 17.6.1].

Theorem 2.8.11. The following holds in the Solovay model derived from ameasurable cardinal. Let E be an analytic equivalence relation on a Polish spaceX. The following are equivalent:

1. E is unpinned;

2. F2 ≤ E or Eω1≤a E.


Proof. Let κ be a measurable cardinal and let W be the derived Solovay model.In W , (2) certainly implies (1) as the proofs that F2, Eω1 are unpinned work inZF, and the proof that pinned equivalence relations persist downwards in thereducibility orderings works in ZF+DC.

For the implication (1)→ (2), assume that W |= E is unpinned. There mustbe a poset P and a nontrivial E-pinned name τ on the poset P , both in W .Both P and τ must be definable in W from a ground model parameter z ∈ 2ω.For simplicity of the notation assume that z ∈ V . Return to V . Let Q be thetwo-step iteration Coll(ω,< κ) ∗ P , and write σ for the Q-name obtained fromthe P -name τ . There are two cases.Case 1. There is a condition q ∈ Q such that the Q � q-name σ is E-pinned. Inthis case, we will conclude that Eω1 ≤a E and use the Shoenfield absoluteness totransfer the almost reducibility to the Solovay model. To simplify the notationassume that q is the largest element of the poset Q.

Observe that the name σ cannot be E-equivalent to any name on a posetof size < κ. Suppose for contradiction that R is a poset of size < κ and χan E-pinned R-name such that 〈R,χ〉 E 〈Q, σ〉. In W , let H ⊂ R be a filtergeneric over V . By Proposition 2.1.5 applied over V , W |= P τ E χ/H. Thiscontradicts the assumption that V [G] |= P ∀x ∈ X ∩ V [G] ¬τ E x.

Now, it follows that κ(E) ≥ κ. By Theorem 2.5.6(2), since κ is a measurablecardinal, κ(E) =∞. By Theorem 2.8.9, Eω1

≤a E as desired.Case 2. For every condition q ∈ Q, the Q � q-name σ is not E-pinned. In thiscase, we will conclude that F2 is reducible to E. Then, a Shoenfield absolutenessargument shows that the Borel reduction of F2 to E remains a Borel reductionalso in the Solovay model.

Fix a countable elementary submodel M of a large structure containing thecode for E, the posets P,Q and the name τ . Write R = Coll(ω,< κ) ∩M . LetZ = 2ω, and use Proposition 1.7.8 to find a Borel function G from Z to P(M)such that for every finite set a ⊂ Z the filters G(z) ⊂ R for z ∈ a are mutuallygeneric over M . For every set a ⊂ Z write 2ωa =

⋃{2ω ∩M [

∏z∈bG(z)] : b ⊂ a

finite}, Ma = M(2ωa ), Pa and τa for the poset and name in Ma defined in themodel M(2ωa ) by the formulas φP and φτ . Similar usage will prevail for functionsy : ω → I, writing Py = Prng(y) etc.

Lemma 2.8.12.

1. whenever a ⊂ Z is a nonempty set then there exists a filter h ⊂ R genericover M such that 2ω ∩M [h] = 2ωa ;

2. whenever a, b, c ⊂ I are pairwise disjoint countable nonempty sets thenthere is a filter ha×hb×hc ⊂ R3 generic over M such that 2ωa = 2ω∩V [ha]and similarly for 2ωb and 2ωc ;

3. whenever a, b are distinct countable subsets of I then Pa ×Pb ¬τa E τb.

Proof. For (1), let S = {k : ∃α ∈ κ ∩M ∃b ⊂ a b is finite and k ⊂ Coll(ω,<α) ∩M is a filter generic over M and k ∈ M [G(z) : z ∈ b]} and order S byinclusion. Let K ⊂ S be a sufficiently generic filter; we claim that h =

⋃K

2.8. DICHOTOMIES 79

works as desired. Indeed, a simple density argument shows that h ⊂ R is anultrafilter all of whose proper initial segments are generic over M . By the κ-c.c.of Coll(ω,< κ), the filter h is in fact generic over M itself. A straightforwardgenericity argument using Fact 1.7.12 then shows that 2ωa = 2ω∩M [h] as desired.

(2) follows easily from (1). Let ha, hb, hc ⊂ R be any filters obtained from(1); we will show that these filters are in fact mutually generic over the modelV . Since Coll(ω,< κ)3 has κ-c.c.c., it is enough to show that for every ordinalα ∈ κ, the filters hαa = ha ∩ Coll(ω,< α), hαb = hb ∩ Coll(ω,< α), and hαc =hc ∩ Coll(ω,< α) are mutually generic over V . Since the filters hαa , hαb andhαc are coded by reals in the models M [ha], M [hb], and M [hc],there are finitesubsets a′, b′, c′ of a, b, c respectively such that hαa ∈M [

∏i∈a′ gi] etc. The mutual

genericity now follows from the general Corollary 1.7.9 about product forcing.(3) is proved in several parallel cases depending on the mutual position of

the sets a, b vis-a-vis inclusion. We will treat the case in which all three setsa ∩ b, a \ b, b \ a are nonempty. Suppose for contradiction that Pa × Pb τa E τb. From (2), it follows that in V , the triple product Coll(ω,< κ)3 forcesP{0,1} × P{1,2} τ{0,1} E τ{1,2}. Then, the quadruple product Coll(ω,< κ)4

forces in V that P{0,1} × P{1,2} × P{2,3} τ{0,1} E τ{1,2} E τ{2,3}, in particular

P{0,1}×P{2,3} τ{0,1}×τ{2,3}. In view of (2) again, this means that the product

Coll(ω,< κ)× Coll(ω,< κ) forces Pleft × Pright τleft E τright. In other words,

(Coll(ω,< κ) ∗ P ) × (Coll(ω,< κ) ∗ P ) forces σleft E σright, contradicting thecase assumption.

Write Y = (2ω)ω = dom(F2). Use the Lusin–Novikov theorem to find a Borelmap g : Y → (2ω)ω such that for every y ∈ Y , g(y) enumerates the set 2ωy . UseLemmas 2.8.12(1) and 2.8.5 to find a Borel map h : Y → P(Q∩M) such that forevery y ∈ Y , h(y) ⊂ Q is a filter generic over M and rng(g(y)) = 2ω ∩ V [h0(y)],where h0(y) ⊂ R is the filter generic over M obtained from h(y). Let k : Y → Xbe given by k(y) = τ/h(y); this is a Borel map by Lemma 2.8.4. We will showthat k is a reduction of F2 to E.

First, assume that y0, y1 ∈ Y are F2-related. Then rng(y0) = rng(y1),rng(g(y0)) = rng(g(y1)), and so My0 = My1 , Py0 = Py1 and τy0 = τy1 . Let H ⊂Py0 be a filter generic over both countable models My0 [k(y0)] and My1 [k(y1)]and let x = τy0/H. By the forcing theorem applied in the model My0 = My1 andthe fact that τy0 is an E-pinned name, conclude that x E k(y0) and x E k(y1),so k(x0) E k(x1) as desired.

Second, assume that y0, y1 ∈ Y are not F2-related. Choose a sufficientlygeneric filter H0×H1 ⊂ Py0 ×Py1 so that H0 is generic over My0 [k(y0)] and H1

is generic over My1 [k(y1)]. As the names τy0 and τy1 are E-pinned, the forcingtheorem in the models My0 and My1 implies that k(y0) E τy0/H0 and k(y1) Eτy1/H1. Now, τy0/H0 E τy1/H1 fails by Lemma 2.8.12(3), so k(y0) E k(y1)must fail as well. This completes the proof.

Corollary 2.8.13. The following holds in the symmetric Solovay model derivedfrom a measurable cardinal. Let E be a Borel equivalence relation on a Polishspace X. E is unpinned if and only if F2 ≤ E.


Proof. This follows from Theorem 2.8.11 once we show that the option Eω1 ≤a Eis not available for any Borel equivalence relation E. This in turn follows easilyfrom results on the pinned cardinal κ(E) obtained in Section 2.5: κ(E) < ∞by Theorem 2.5.6(1), κ(Eω1

) = ∞ by Example 2.4.6, and the pinned cardinalis monotone with respect to the reducibility ordering ≤a–Theorem 2.5.4.

Chapter 3

Turbulence

In this chapter, we investigate pairs of generic extensions V [H0] and V [H1] suchthat V [H0]∩V [H1] = V . In a particularly successful application, this allows usto restate Hjorth’s concept of turbulence in geometric terms and produce manygeneralizations of the associated ergodicity theorem [52, Theorem 12.5].

3.1 Independent functions

The purpose of this section is to find a practical way of producing many exten-sions which satisfy the condition V [H0] ∩ V [H1] = V without being mutuallygeneric. We need a folkloric observation. Recall that for a Polish space X, theposet PX consists of nonempty open subsets of X ordered by inclusion; xgen isits name for the unique element of X belonging to all open sets in the genericfilter.

Proposition 3.1.1. Let X,Y be Polish spaces and f : X → Y be a continuousopen function. Then PX forces f(xgen) to be a PY -generic element of Y .

Proof. It is just necessary to show that for every open dense set D ⊂ Y , PX f(xgen) ∈ D holds. To this end, let O ∈ PX be a condition. The set f ′′O ⊂ Y isopen; therefore, it has nonempty intersection with D. Consider the nonemptyopen set O′ = (f−1(f ′′O ∩ D)) ∩ O ⊂ O. For every point x ∈ O′, f(x) ∈ Dholds, so O′ f(xgen) ∈ D as required.

Now suppose that X,Y0, Y1 are Polish spaces and f0 : X → Y0 and f1 : X → Y1

are continuous open maps. We want to find a criterion implying that PX V [f0(xgen)] ∩ V [f1(xgen)] = V . The following turbulence–like definition is cen-tral.

Definition 3.1.2. Let X,Y0, Y1 be Polish spaces and f0 : X → Y0 and f1 : X →Y1 be continuous open maps. Let k ∈ ω be a number. A walk (of points) oflength k is a sequence 〈xi : i ≤ k〉 of points in X such that for each i ∈ k, eitherf0(xi) = f0(xi+1) or f1(xi) = f1(xi+1) holds.

81

82 CHAPTER 3. TURBULENCE

Definition 3.1.3. Let X,Y0, Y1 be Polish spaces and f0 : X → Y0 and f1 : X →Y1 be continuous open maps. The functions f0, f1 are independent if for everynonempty open set O ⊂ X there is a nonempty open set A ⊂ Y0 such that forall nonempty open subsets B0, B1 ⊂ A there is a walk consisting of points in Ostarting in f−1

0 B0 and ending in f−10 B1.

It may appear that the definition is not symmetric with respect to the mapsf0, f1. In fact, the definition is symmetric, and this follows as a small corollaryfrom the following central theorem.

Theorem 3.1.4. Suppose that X,Y0, Y1 are Polish spaces and f0 : X → Y0 andf1 : X → Y1 are continuous open maps. The following are equivalent:

1. f0, f1 are independent;

2. PX V [f0(xgen)] ∩ V [f1(xgen)] = V where xgen is the PX-name for thegeneric element of X.

Proof. For the (1)→(2) implication, we need a preliminary definition and aclaim. Let k ∈ ω be a number. A walk of open sets of length k is a sequence〈Oi : i ≤ k〉 of nonempty open subsets of X such that for each i ∈ k, eitherf ′′0 Oi = f ′′0 Oi+1 or f ′′1 Oi = f ′′1 Oi+1.

Claim 3.1.5. Let k ∈ ω be a number. Suppose that 〈xi : i ≤ k〉 is a walk ofpoints, Oi ⊂ X are open sets such that xi ∈ Oi, and D is a finite collectionof open subsets of X. Then there is a walk 〈Pi : i ≤ k〉 of open sets such thatPi ⊂ Oi holds for each i ≤ k and for each U ∈ D and each i ≤ k, Pi is eitherdisjoint from U or a subset of U .

Proof. By induction on k. The case k = 0 is trivial. Suppose that the claimhas been proved for k and 〈xi : i ≤ k + 1〉 is a walk of points, Oi are open sets,and D is a finite collection of open subsets of X. Suppose for definiteness thatf0(xk) = f0(xk+1). Shrinking Ok and Ok+1 if necessary, we may assume thatf ′′0 Ok = f ′′0 Ok+1. For each open set U ∈ D let U ′ =

⋃{V ⊂ X : V is open and

f−10 f ′′0 V ⊂ U} and let E = {U ′ : U ∈ D}. By the induction hypothesis, there is

a walk 〈Pi : i ≤ k〉 of open sets such that Pi ⊂ Oi and Pi is either disjoint fromor a subset of each element of D ∪E. Let Pk+1 = f−1

0 f ′′0 Pk ∩Ok+1 and observethat the walk 〈Pi : i ≤ k + 1〉 of open sets works as required.

Assume now that the functions f0, f1 are independent. Since the models in (2)satisfy ZFC, it is enough to show that every set of ordinals in the intersection isactually in V . To this end, suppose that τ0, τ1 are PY0

and PY1-names for sets of

ordinals. We will abuse notation somewhat and write τ0 also for the PX -namefor τ0/f0(xgen) and τ1 for the PX -name for τ1/f1(xgen). Suppose that O ⊂ Xis a nonempty open set forcing τ0 = τ1. We must find a nonempty open setO′ ⊂ O deciding the statement α ∈ τ0 for every ordinal α.

Let A ⊂ Y0 be a nonempty open set standing witness to the independenceof the functions f0, f1. We claim that for every ordinal α, in the poset PY0

,

3.1. INDEPENDENT FUNCTIONS 83

A α ∈ τ0 or A α /∈ τ0. Once this is proved, let O′ = O ∩ f−10 A and note

that O′ decides the membership of every ordinal in τ0 as desired.

Suppose towards a contradiction that there exist an ordinal α and nonemptyopen sets B0, B1 ⊂ A such that B0 α ∈ τ0 and B1 α /∈ τ0. Let 〈xi : i ≤ k〉be a walk of points in O such that f0(x0) ∈ B0 and f0(xk) ∈ B1. Let U0 =⋃{U : U ⊂ X and U α ∈ τ0} and U1 =

⋃{U : U ⊂ X and U α ∈ τ1}. By

the claim, there must be a walk of open sets 〈Oi : i ≤ k〉 such that each set Oiis a subset of O, O0 ⊂ f−1

0 B0, Ok ⊂ f−10 B1, and each Oi is either disjoint from

or a subset of U0, and either disjoint from or a subset of U1. This means inparticular that each Oi decides both statements α ∈ τ0 and α ∈ τ1.

Now, since O0 α ∈ τ0 and Ok α /∈ τ0, there must be a number i ∈ ksuch that Oi α ∈ τ0 and Oi+1 α /∈ τ0. It then must be the case thatf ′′0 Oi, f

′′0 Oi+1 ⊂ Y0 are disjoint; therefore, the sets f ′′1 Oi, f

′′1 Oi+1 ⊂ Y1 are equal.

This means that the sets Oi, Oi+1 must either both force α ∈ τ1 or both forceα /∈ τ1. In conclusion, one of the sets Oi, Oi+1 forces τ0 6= τ1, a contradiction.

To see why the negation of (1) implies the negation of (2), let O ⊂ X be anonempty open set witnessing the failure of (1). For each y ∈ Y0, define the orbitof y to be the set of all elements z ∈ Y0 such that there is a walk 〈xi : i ≤ 2k〉 inO such that f0(x0) = y and f0(x2k) = z. Let x ∈ O be a PX -generic and workin the model V [x].

Claim 3.1.6. The orbit of f0(x) is nowhere dense in Y0.

Proof. If the orbit were dense in some nonempty open set B ⊂ Y0, then in theground model, the set B would show that O is not a witness to the failure of (1).To see this, suppose that B0, B1 ⊂ B are nonempty open subsets of B. Workingin V [x], find walks w0, w1 in O which lead from f0(x) to B0 and B1, invert w0

and concatenate with w1 and get a walk from B0 to B1. By the Mostowskiabsoluteness between V [x] and V , there must be such a walk in V as well.

Now look at the set A = {B ⊂ Y0 : B is a basic open subset of Y0 and B containsno element of the orbit of f0(x)}.

Claim 3.1.7. A ∈ V [f0(x)] ∩ V [f1(x)].

Proof. In V [x], the set A is defined as the set of all basic open sets B ⊂ Y0

such that there is no walk 〈xi : i ≤ 2k〉 in O such that f0(x0) = f0(x) andf0(x2k) ∈ B. This is a Π1

1 definition with parameter f0(x) which thereforeyields the same set in the model V [f0(x)].

In V [x], the same set A also has an alternative definition: it is the set ofall basic open sets B ⊂ Y0 such that there is no walk 〈xi : i ≤ 2k〉 in O suchthat f1(x0) = f1(x) and f0(x2k) ∈ B. To see this, note that a walk withf0(x0) = f0(x) can be transformed into a walk with f1(x0) = f1(x) (and alsovice versa) by simply adding the point x twice to the beginning of the walk.This alternative definition of the set A is again Π1

1 with parameter f1(x), soMostowski absoluteness yields the same set in the model V [f1(x)].


To show that (2) fails, it is enough to argue that A /∈ V . However, if A ∈ V then⋃A ∈ V is an open dense subset of Y0 by Claim 3.1.6, and f0(x) /∈

⋃A. This

contradicts Proposition 3.1.1: the point f0(x) ∈ Y is forced to be PY -genericover the ground model, so to belong to every open dense set in the groundmodel.

3.2 Examples

Several groups of results in this book depend on an identification of a suitablepair of independent maps. The most important example comes from Hjorth’snotion of turbulence of group actions. Recall the standard definition:

Definition 3.2.1. [48, Section 13.1] Let Γ be a Polish group continuously actingon a Polish space Y .

1. If U ⊂ Γ, O ⊂ Y are sets then a U,O-walk is a sequence 〈yi : i ≤ k, γi : i <k〉 such that for each i ≤ k yi ∈ O holds, and for each i < k γi ∈ U andyi+1 = γi · yi both hold;

2. if y ∈ O then U,O-orbit of y is the set of all z ∈ O such that there is aU,O-walk starting at y and ending at z;

3. the action is turbulent at y ∈ Y if for all open sets U ⊂ Γ and O ⊂ Xwith 1 ∈ U and y ∈ O the U,O-orbit of y is somewhere dense;

4. the action is generically turbulent if its orbits are meager and dense andthe set of points of turbulence is comeager.

Now, suppose that Γ is a Polish group acting continuously on a Polish space Y .Let X = {〈γ, y0, y1〉 ∈ Γ×Y ×Y : γ ·y0 = y1}; this is a closed subset of Γ×Y ×Yand Polish in the inherited topology. Let f0 : X → Y be the projection into thesecond coordinate, f1 : X → Y be the projection into the third coordinate, andlet f2 : X → Γ×Y be the projection into the first two coordinates. Since the setX can be viewed as a graph of a continuous function of any pair of coordinatesinto the remaining one, these mappings are continuous and open. With thisnotation in hand, we prove:

Theorem 3.2.2. Let Γ be a Polish group continuously acting on a Polish spaceY such that all orbits are meager and dense. The following are equivalent:

1. the action is generically turbulent;

2. f0, f1 : X → Y is an independent pair of functions;

3. PΓ × PY forces V [xgen ] ∩ V [γgen · xgen ] = V .

Proof. For the implication (1)→(2), suppose that the action is generically tur-bulent. Let O ⊂ X be a nonempty open set. Find a point δ ∈ Γ and open sets

3.2. EXAMPLES 85

U ⊂ Γ and O0, O1 ⊂ Y such that 1 ∈ U , U = U−1, and (δ ·U×O0×O1)∩X ⊂ O.Find a point y ∈ O0 such that the U,O0-orbit of y is somewhere dense; let A ⊂ Ybe a nonempty open set such that the U,O0-orbit of y is dense in A. We claimthat A ⊂ Y is an open set witnessing the independence of functions f0 and f1.

To see this, let B0, B1 ⊂ A be nonempty open sets. Concatenating U,O0-walks from y to B0 and B1, we can find a U,O0-walk 〈yi : i ≤ k, γi : i < k〉 whichstarts in B0 and ends in B1. Consider the sequence 〈xi : i ≤ 2k〉 of points inthe open set O ⊂ X given by x2i = 〈δ, yi, δ · yi〉 and x2i+1 = 〈δγi, yi, δyi+1〉 ifi < k. It is immediate that this is an walk of points in the set O in the sense ofDefinition 3.1.2, confirming the independence of functions f0, f1.

For the implication (2)→(1), suppose that the functions f0, f1 are indepen-dent. Let U ⊂ Γ be an open neighborhood of 1 and O0 ⊂ Y be a nonempty openset. Choose an open neighborhood U0 ⊂ Γ of the unit such that U−1

0 · U0 ⊂ Uand let O = {〈γ, y0, y1〉 ∈ X : γ ∈ U0 and y0, y1 ∈ O0}. Suppose that 〈xi : i ≤ k〉is a walk of points in the set O in the sense of Definition 3.1.2 and xi = 〈δi, zi, z′i〉;then the sequence 〈zi : i ≤ k, γi : i < k〉 is an U,O0-walk in the sense of Defini-tion 3.2.1, where γi = 1 if zi = zi+1 and γi = δ−1

i+1δi if zi 6= zi+1 (in which casez′i = z′i+1).

Now, if A ⊂ O0 is an open set witnessing the independence of f0, f1 for Oand B ⊂ A is a nonempty open set, then the set {y ∈ A : there is a walk ofpoints in O from y to B} is open dense in A, and by the previous paragraph it isa subset of the set {y ∈ A : there is an U,O0-walk from y to B}. Thus, the setCU,O0

= {y ∈ Y : either y /∈ A or the U,O0-orbit of y is dense in A} is comeager.The set of turbulent points then contains the intersection

⋂{CU,O0 : U ⊂ Γ is a

basic open neighborhood of 1 and O0 ⊂ Y is a basic open set} and is comeager.The equivalence (2)↔(3) follows from Theorem 3.1.4 and the fact that the

function f2 is continuous and open, so the first two coordinates of the PX -genericpoint are generic for PΓ × PY by Proposition 3.1.1.

A somewhat similar characterization theorem can be proved for analyticideals on ω.

Definition 3.2.3. [13] A set I ⊂ P(ω) is ω-hitting if for every countable collec-tion of infinite subsets of ω there is a set b ∈ I which has nonempty intersectionwith each set in the collection.

A typical analytic ω-hitting ideal is Ip, where p is a partition of ω into finite setsand Ip is the collection of all sets a ⊂ ω such that for some number n ∈ ω, forevery set b ∈ p, |a ∩ b| ≤ n. In fact, [43] shows that every Borel ω-hitting idealcontains one of the ideals Ip as a subset in a suitable sense. ω-hitting ideals giverise to pairs of independent functions in the proof of the following theorem:

Theorem 3.2.4. Let I be an analytic ideal on ω. The following are equivalent:

1. I is ω-hitting;

2. in some forcing extension, there are points y0, y1 ∈ 2ω separately Cohen-generic over V such that y0 =I y1 and V [y0] ∩ V [y1] = V .


Proof. For the direction (1)→(2), assume that the ideal I is ω-hitting. Leth : ωω → P(ω) be a continuous map such that I = rng(h). Let T = {t ∈ω<ω : h′′[t] is ω-hitting}. The set T ⊂ ω<ω is clearly closed under initial segment.

Claim 3.2.5. For all t ∈ T there is n ∈ ω such that for all m > n there isz ∈ [T � t] such that m ∈ h(z).

Proof. Suppose towards a contradiction that the conclusion fails for some nodet ∈ T . Let a = ω \

⋃h′′[T � t]; the set a ⊂ ω is infinite. For each s ∈ ω<ω \ T

let cs be a countable collection of infinite subsets of ω such that for all z ∈ [s],h(z) has finite intersection with one element of cs. Let c =

⋃s/∈T cs ∪ {a}; it

is not difficult to see that for every z ∈ [t], h(z) has finite intersection with anelement of c, contradicting the assumption that t ∈ T .

Now, let X be the closed set of all triples 〈y0, y1, z〉 ⊂ 2ω × 2ω × [T ] suchthat {n ∈ ω : y0(n) 6= y1(n)} ⊂ h(z). Let f0, f1 : X → 2ω be the projectionsinto the first and second coordinate. It is not difficult to see that both of thesemaps are continuous and open. In view of Theorem 3.1.4, the following claimconcludes the proof of (2).

Claim 3.2.6. The maps f0, f1 are independent.

Proof. Let O ⊂ X be an open set. Find u0, u1 ∈ 2<ω and t ∈ T such that([u0]× [u1]× [t])∩X is a nonempty subset of O. Thinning out further, we mayassume that the binary strings u0, u1 have the same length and for all z ∈ [T � t]and every l such that u0(l) 6= u1(l), n ∈ h(z) holds. Let n ∈ ω be such that forall m > n there is z ∈ [T � t] such that m ∈ h(z). Let v ∈ 2<ω be any binarystring extending u0 of length > n and A = [v]. We claim that the open set Awitnesses the definition of independence for f0, f1 and O.

Indeed, suppose that w0, w1 are two binary strings of the same length ex-tending v. To produce a walk from [w0] to [w1], list all entries on which thestrings w0, w1 differ by {mj : j ∈ i}. Let y ∈ [w0] be any point, and let yj ∈ 2ω

be the point obtained from y by flipping the outputs at all mk for k ∈ j. Thus,yj differs from yj+1 only at entry mj and yi ∈ [w1]. Let y′j ∈ 2ω be the pointobtained by replacing the initial segment u0 ⊂ yj with u1. Also, let zj ∈ [T � t]be any point such that mj ∈ h(zj). Now, define points x2j+1, x2j+2 ∈ X forj ∈ i as follows:

• x0 = 〈y0, y′0, z〉 for any z ∈ [T � t];

• x2j+1 = 〈yj , y′j+1, zj〉;

• x2j+2 = 〈yj+1, y′j+1, z〉 for any z ∈ [T � t].

Clearly, the sequence 〈xk : k ≤ 2i〉 is a walk from [w0] to [w1] as required.

To prove that (2) implies (1), assume that (1) fails, as witnessed by a count-able collection {an : n ∈ ω} of infinite subsets of ω. Suppose that in some genericextension, y0, y1 ∈ 2ω are =I -related points which are separately Cohen-generic

3.2. EXAMPLES 87

over V . The set b = {n ∈ ω : y0(n) 6= y1(n)} belongs to the ideal I; therefore,there is a number n ∈ ω such that b ∩ an is finite. It follows that the func-tions y0 � an and y1 � an are modulo finite equal; therefore, they belong toV [y0] ∩ V [y1]. By a genericity argument, y0 � an /∈ V holds, so (2) fails.

Another example of an independent pair of functions deals with Polish fields.

Theorem 3.2.7. Let Φ be an uncountable Polish field. Let X = {〈u0, u1〉 ∈(Φ2)2 : u0 ·u1 = 1} with the topology inherited from Φ4. The projection functionsfrom X to the first and the second coordinate are independent.

Proof. Write f0 : X → Φ2 and f1 : X → Φ2 for the two projection functions. Itis not difficult to see that the maps f0, f1 are continuous and open. Towardsthe independence, let O ⊂ (Φ2)2 be an open set with nonempty intersectionwith X. Let 〈u0, u1〉 be a point in X ∩ O. Let v0 ∈ Φ2 be any nonzero vectorsuch that u0 · v0 = 0. By the continuity of multiplication, there must be anonzero field element y ∈ Φ such that the pair 〈u0, u1 + yv0〉 belongs to O; italso belongs to X. The determinant of the 2×2 matrix consisting of the vectorsu1 and v1 = u1 + yv0 is nonzero. It follows that there are open neighborhoodsO0, O1 ⊂ Φ2 of u1, v1 respectively such that for each choice of w0 ∈ O0 andw1 ∈ O1, the system of equations x · w0 = 1 and x · w1 = 1 has a uniquesolution, and the solution x ∈ Φ2 is moreover such that 〈x,w0〉, 〈x,w1〉 are bothelements of O.

We claim that the set O0 ⊂ Φ2 stands witness to the independence of thefunctions f0, f1. Indeed, if w0, w

′0 ∈ O0 are arbitrary points, then choose any

point w1 ∈ O1, find vectors x0 and x1 such that x0 · w0 = 1, x0 · w1 = 1,x1 · w′0 = 1 and x1 · w1 = 0, and observe that 〈x0, w0〉, 〈x0, w1〉, 〈x1, w1〉,,〈x1, w

′0〉 is a walk of points in O starting at w0 and ending at w′0.

The following theorem is closely related to the previous one in view of thefact that its statement is intended to be applied with algebraic curves.

Theorem 3.2.8. Let C ⊂ R be a smooth curve containing no straight segments,which is closed as a subset of R2. Let X = 〈u0, u1〉 ∈ R2 : u0 − u1 ∈ C}. Theprojection functions from X to the first and second coordinate are independent.

Proof. Since the set C is closed, X ⊂ (R2)2 is closed and equipped with thetopology inherited from (R2)2. Let f0 : X → R2 and f1 : X → R2 be theprojection functions. Clearly, f0 and f1 are continuous. They are also open: ifO ⊂ (R2)2 is an open set, 〈u0, u1〉 ∈ X∩O is a point, and ε > 0 is a real numbersuch that for each vector v ∈ R2 of norm smaller than ε the point 〈u0 +v, u1 +v〉belongs to O, then the open neighborhood {u0 + v : |v| < ε} of u0 is a subset ofthe f0-image of X ∩O, as the points 〈u0 + v, u1 + v〉 belong not only to O butalso to X as u0 − u1 = (u0 + v)− (u1 + v).

To see the independence, fix an open set O ⊂ (R2)2 and a point 〈u0, u1〉 ∈X ∩O. Translating if necessary, we may assume that u1 = 0. Find an open ballB ⊂ R2 close to u0 such that B × (B − B) ⊂ O and C ∩ B is a simple curveconnecting two distinct points at the boundary of B. As the curve C contains


no straight segments, there must be points p0, p1 ∈ B∩C such that the tangentvectors at p0 and p1 are nonzero and distinct. An inspection reveals that theremust be an open set P ⊂ R2 close to 0 such that p0 − P ⊂ B and for everyvector v ∈ P , the points p0 + v and p1 + v end up on the opposite sides of thecurve C and in the ball B, and the v-shift of the part of the curve C betweenthe points p0, p1 is a subset of B. By the Jordan curve theorem, the curves Cand C + v must intersect in the ball B.

We claim that the set p0 − P ⊂ R2 witnesses the independence of the mapsf0, f1. Let v0, v1 ∈ P be two vectors and q0, q1 ∈ B be respective points in theintersections C∩ (C+v0) and C∩ (C+v1). Then 〈p0−v0, p0−q0〉, 〈p0, p0−q0〉,〈p0, p0 − q1〉, and 〈p0 − v1, p0 − q1〉 constitutes a walk from p0 − v0 to p0 − v1

using points in O.

The assumption that C contains no straight segments is necessary in Theo-rem 3.2.8. Suppose that C is a line passing through the origin and argue thatthe projection functions are not independent. To see this, consider the PX -generic pair 〈u0, u1〉. The line in the direction of C through u0 passes throughu1 and vice versa. Let B be the collection of all basic open subsets of R2 whichhave empty intersection with this line; thus, B ∈ V [u0] ∩ V [u1]. The set

⋃B

is open dense in R2 and contains neither u0 nor u1. Since u0, u1 are Cohen-generic points, they meet all open dense subsets of R2 coded in the groundmodel; it follows that B /∈ V . By Theorem 3.1.4, the projection functions arenot independent.

3.3 Placid equivalence relations

The characterization of turbulence in Theorem 3.2.2 leads to many Borel nonre-ducibility results and cardinal preservation results in the generic extensions ofthe Solovay model. To state these results succinctly, the following definitionsare helpful.

Definition 3.3.1. Let E be an analytic equivalence relation on a Polish spaceX. We say that E is placid if, whenever V [H0] and V [H1] are separately genericextensions of V (inside some ambient generic extension) such that V [H0] ∩V [H1] = V and x0 ∈ V [H0] and x1 ∈ V [H1] are E-related points in the spaceX, then they are E-related to some point in V .

Definition 3.3.2. Let E be an analytic equivalence relation on a Polish spaceX. We say that E is virtually placid if, whenever V [H0] and V [H1] are sepa-rately generic extensions of V (inside some ambient generic extension) such thatV [H0] ∩ V [H1] = V and 〈Q0, τ0〉 ∈ V [H0] and 〈Q1, τ1〉 ∈ V [H1] are E-relatedE-pins, then they are E-related to some E-pin in V .

In other words, an equivalence relation E is placid if disjoint generic extensionsdo not share any E-classes which are not represented in V . An equivalencerelation E is virtually placid if disjoint generic extensions do not share any

3.3. PLACID EQUIVALENCE RELATIONS 89

virtual E-classes which are not represented in V . The following propositionsencapsulate the basic properties of the two classes of equivalence relations.

Proposition 3.3.3. Let E be an analytic equivalence relation on a Polish spaceX. E is virtually placid if and only if for any separately generic extensionsV [H0], V [H1] such that V [H0] ∩ V [H1] = V and E-related points x0 ∈ V [H0]and x1 ∈ V [H1], x0 and x1 are realizations of some virtual E-class in V .

Proof. The left-to-right implication is immediate from the definitions. For theright-to-left implication, suppose that V [H0] and V [H1] are separately genericextensions such that V [H0] ∩ V [H1] = V and 〈Q, τ0〉 ∈ V [H0] and 〈Q1, τ1〉 ∈V [H1] are E-related pins. Let K0 ⊂ Q0 and K1 ⊂ Q1 be filters mutually genericover the model V [H0, H1] and let x0 = τ0/K0 and x1 = τ1/K1. By the mutualgenericity, V [H0][K0] ∩ V [H1][K1] = V , and the assumption on E gives thatx0, x1 are realizations of some virtual E-class from the ground model. It followsthat both E-pins are E-related to some E-pin from V .

Proposition 3.3.4. An analytic equivalence relation E on a Polish space X isplacid if and only if it is pinned and virtually placid.

Proof. The right-to-left implication is immediate. If E is a pinned and virtuallyplacid equivalence relation and V [H0], V [H1] are separately generic extensionssuch that V [H0]∩V [H1] = V , then any E-class represented in both must be rep-resented as a virtual class in V by the virtual placid assumption, and this virtualclass must be trivial by the pinned assumption. This means that E is placid.For the left-to-right implication, if E is placid then it must be pinned becauseif V [H0], V [H1] are mutually generic extensions then V [H0] ∩ V [H1] = V bythe product forcing theorem. To show that E must be virtually placid, supposethat V [H0], V [H1] are separately generic extensions. By Proposition 3.3.3, it isenough to verify that every E-class represented in both extensions is a realiza-tion of some virtual E-class in V . However, the placid assumption even impliesthat it is a realization of a trivial virtual E-class in V .

A good example of an equivalence relation which is virtually placid but notplacid is F2. It is not placid since it is not pinned.

The following ergodicity theorem is a great generalization of the ergodicitytheorems of Hjorth and Kechris [52, Theorem 12.5], and it is the main motiva-tion behind the definition of placid and virtually placid classes of equivalencerelations.

Theorem 3.3.5. Suppose that a Polish group Γ acts continuously and in agenerically turbulent way on a Polish space X such that the resulting orbit equiv-alence relation E is analytic, with all orbits dense. Suppose that F is a virtuallyplacid equivalence relation on a Polish space Y . Suppose that h : X → Y isa homomorphism of E to F . Then there is a comeager set B ⊂ X which ismapped into a single F -class.

Proof. Let γ ∈ Γ, x ∈ X be a mutually generic pair of points for the posets PΓ

and PX . Theorem 3.2.2 implies the instrumental equality: V [x] ∩ V [γ · x] = V .


Since x E γ · x holds, it must be the case that h(x) F h(γ · x) must hold. Bythe virtual placidity of the equivalence relation F , it must be the case that h(x)is a realization of a ground model virtual F -class. However, the poset PX iscountable, so all virtual classes realized in its extension are already representedin the ground model by Theorem 2.6.2. Thus, there must be a point y ∈ Y ∩ Vsuch that h(x) F y. Since the generic point x ∈ X avoids all ground model codedmeager sets, it must be the case that the set B = h−1([y]E) ⊂ X is nonmeager.Since this is an analytic set which is invariant under the continuous group actionall of whose orbits are dense, it follows that the set B is comeager.

3.4 Examples and operations

The extent of the classes of placid and virtually placid equivalence relations isbest explored using the following closure theorems.

Theorem 3.4.1. The class of placid equivalence relations is closed under thefollowing operations:

1. Borel almost reduction;

2. countable product;

3. countable increasing union;

4. countable factor;

The class of virtually placid equivalence relations is closed under the same op-erations and the Friedman–Stanley jump.

Proof. In view of Proposition 3.3.4, it is enough to show the closure of virtualplacidity under these operations, since the class of pinned equivalence relationsis closed under (1–4) by the work of Chapter 2. For virtual placidity, we willprove (1) and the closure under the Friedman–Stanley jump.

For (1), suppose that E,F are analytic equivalence relations on Polish spacesX,Y and h : X → Y is a Borel function which is a reduction of E to Y every-where except for a set Z ⊂ X consisting of countably many E-classes. Supposethat F is virtually placid and work towards the conclusion that E is virtuallyplacid. Let V [H0], V [H1] be generic extensions such that V [H0] ∩ V [H1] = Vand let x0 ∈ V [H0] and x1 ∈ V [H1] be E-related points in X; we need to showthat they are realizations of some virtual E-class in V . If x0 ∈ Z then thiscertainly occurs as V contains a countable set of points whose E-classes coverZ, and this feature of the countable set persists to V [H0] by a Shoenfield abso-luteness argument. If x0 /∈ Z then x1 /∈ Z; look at the points h(x0) ∈ V [H0] andh(x1) ∈ V [H1]. These are F -related points; since F is placid they realize somevirtual F -class represented by some F -pin 〈Q, σ〉. By a Shoenfield absolutenessargument, Q ∃x ∈ X \Z h(x) E σ; let τ be a Q-name for such a point x ∈ X.It is not difficult to see that 〈Q, τ〉 is an E-pin and x0, x1 realize the virtualclass of 〈Q, σ〉.

3.4. EXAMPLES AND OPERATIONS 91

For the Friedman–Stanley jump, suppose that E is a virtually placid equiv-alence relation on a Polish space X, V [H0] and V [H1] are separately genericextensions of V such that V [H0] ∩ V [H1] = V and y0, y1 ∈ Xω are elementsin the respective models such that [rng(y0)]E = [rng(y1)]E . Since E is virtu-ally placid, every element of rng(y0) is a realization of a virtual E-class fromV . Let B be the set of all virtual E-classes in V which have realizations inrng(y0). Since B is also the set of all virtual E-classes in V which have real-izations in rng(y1), it is clear that B ∈ V [H0] ∩ V [H1] = V . Thus, the pointsy0, y1 ∈ Xω are realizations of the virtual E+-class represented by the set B.By Proposition 3.3.3, we conclude that E+ is virtually placid.

Theorem 3.4.2. Suppose that Γ is a Polish group continuously acting on aPolish space X and E is the resulting orbit equivalence relation. The followingare equivalent:

1. E is virtually placid;

2. for every Borel set B ⊂ X such that E � B is Borel, E � B is virtuallyplacid.

Proof. (1) immediately implies (2) since E � B is reducible to E by the identitymap on B. Now suppose that (1) fails, and let P be a poset and τ be a namefor an element of X such that P × P τleft E τright fails, and let Q be a

poset with names H0, H1 for filters generic over the ground model such thatQ V [H0] ∩ V [H1] = V and τ/H0 = τ/H1.

Let M be a countable elementary submodel of a large structure containingall objects mentioned so far. Let M be the transitive collapse of M , and P , Qetc. be the images of P,Q etc. under the transitive collapse map. By Proposi-tion 3.5.5, Q V [H0] ∩ V [H1] = V holds even in V (as opposed to M). LetB = {x ∈ X : ∃H ⊂ P generic over M such that x = τ /H}. We have justproved that E � B is not virtually placid. The proof is completed by a referenceto Corollary 2.8.8–the set B ⊂ X is Borel and the equivalence relation E � B isBorel as well.

The theorem allows the transfer of the virtual placid property from Borelfragments of a given orbit equivalence relation to the whole equivalence relation.Consider the following attractive corollary:

Corollary 3.4.3. Every equivalence relation classifiable by countable structuresis virtually placid.

Proof. In view of Theorem 3.4.1(1), it is only necessary to show that every orbitequivalence relation E obtained from an action of S∞ is virtually placid. ByTheorem 3.4.2, it is only necessary to show that all Borel fragments of E arevirtually placid. To this end, consider the transfinite sequence 〈Fα : α ∈ ω1〉obtained from F1 =identity on 2ω by repeated application of Friedman–Stanleyjump, at limit stages taking disjoint unions. It is well-known [48, Theorem12.5.2] that every Borel equivalence relation classifiable by countable structures


is Borel reducible to Fα for some countable ordinal α. Theorem 3.4.1 iteratedtransfinitely shows that each Fα is virtually placid. Thus, every Borel fragmentof E is virtually placid, and so is E.

There are many Borel placid equivalence relations which cannot be obtainedfrom the identity by repeated application of the operations indicated in The-orem 3.4.1. The following examples deal with equivalence relations associatedwith ideals on countable sets. If a is a countable set and J is an ideal on a,write =J for the equivalence relation on the space X = (2ω)a connecting pointsx0, x1 if the set {n ∈ a : x0(n) 6= x1(n)} belongs to J .

Example 3.4.4. Let a = 2<ω and let J be the branch ideal : the ideal generatedby the subsets of a linearly ordered by inclusion. The equivalence relation =J

is placid.

Proof. Write X = (2ω)2<ω . For every node t ∈ ω<ω write [t] for the set of allnodes in 2<ω extending t. Let V [G0], V [G1] be two generic extensions containingrespective =J -related points x0, x1 ∈ X. Assume that V [G0] ∩ V [G1] = V andwork to find a ground model point x ∈ X which is =J -related to both x0, x1.

Let T = {t ∈ 2<ω : x0 � [t] is not =J -equivalent to any point in the groundmodel}. This is a subtree of ω<ω. If 0 /∈ T then the proof is complete; thus, itis only necessary to derive a contradiction from the assumption 0 ∈ T . First,observe that the tree T cannot have any terminal nodes: if t were a terminalnode of T then one could combine the ground model witnesses for ta0 /∈ T andta1 /∈ T to find a witness for t /∈ T . Second, observe that the definition of thetree T depends only on the =J -class of x0, so T ∈ V [G0] ∩ V [G1] = V . SinceT is a nonempty ground model tree without endnodes, it must have an infinitebranch y ∈ 2ω in the ground model. Since x0 =J x1, there is a number n ∈ ωsuch that for every t ∈ 2<ω such that x0(t) 6= x1(t), either t is incompatiblewith y � n or else t is an initial segment of y. Let e = [y � n] \ {y � m : m ≥ n}and observe that e ∈ V and x0 and x1 coincide on e, therefore x0 � e ∈ V . Ifz ∈ V is any function in 2[y�n] extending x0 � e, then z =J x0 � [t], contradictingthe assumption that y � n ∈ T .

A whole class of ideals with placid equivalence relations is isolated in the fol-lowing definition.

Definition 3.4.5. An ideal I on ω is countably separated if there is a countablecollection A of subsets of ω such that for every b ∈ I and c /∈ I there is a ∈ Asuch that b ∩ a = 0 and c ∩ a /∈ I.

Countably separated ideals include for example the ideal of nowhere dense sub-sets of the rationals; a characterization theorem is proved in [62]. Note that ifI is analytic then the instrumental property of the set A is coanalytic, and byMostowski absoluteness it will hold in all forcing extensions.

Theorem 3.4.6. The equivalence relation =I is placid for every countably sep-arated Borel ideal I on ω.

3.4. EXAMPLES AND OPERATIONS 93

Proof. Let {an : n ∈ ω} be a countable collection of subsets of ω witnessing thecountable separation of the ideal I. Write X = (2ω)ω. Let V [G0], V [G1] begeneric extensions containing respective =I -related points x0, x1 ∈ X. AssumeV [G0] ∩ V [G1] = V and work to find a ground model point x ∈ X which is=I -related to both x0, x1.

Consider the set b = {n ∈ ω : x0 � an is I-equivalent to some element of theground model}. The definition of the set b depends only on the =I -equivalenceclass of x0, therefore the set b belongs to both V [G0] and V [G1], and by theinitial assumptions, to V . Let f be the map with domain b which to each n ∈ bidentifies the =I -class in V which contains x0 � an. Again, the definition off depends only on the =I -class of x0, so f ∈ V [G0] and f ∈ V [G1], thereforef ∈ V . By the Mostowski absoluteness between V and V [G0], there is a pointx ∈ X such that for all n ∈ b, x � an ∈ f(n). We will show that x =I x0 holds.

Suppose towards a contradiction that this fails. Consider the set c = {i ∈ω : x(i) 6= x0(i)} /∈ I and the set d = {i ∈ ω : x0(i) 6= x1(i)} ∈ I. By thecountable separation of the ideal I, there must be a number n ∈ ω such thatc ∩ an /∈ I and d ∩ an = 0. However, the latter equality shows that x0 � an =x1 � an, so x0 � an ∈ V and n ∈ b. But then {i ∈ an : x(i) 6= x0(i)} ∈ I by thechoice of the point x. This contradicts the fact that c ∩ an /∈ I. The proof ofthe theorem is complete.

There is a natural operation on analytic ideals which seeks a countably separatedclosure. For the discussion below, let {an : n ∈ ω} be a fixed countable collectionof subsets of ω.

Definition 3.4.7. Let I be an ideal on ω. I+ is the ideal on ω consisting of allsets b ⊂ ω such that there is c ∈ I such that ∀n ∈ ω b ∩ an /∈ I → c ∩ an 6= 0.

We will start with some simple observations. It is clear that I+ is an idealcontaining I as a subset. If I is analytic, then so is I+. Moreover, I is separatedby the sequence {an : n ∈ ω} just in case I = I+. We will show that undersuitable additional assumption, the jump preserves placidity of the equivalencerelation =I .

Definition 3.4.8. Let I be an ideal on ω. I is skew if for every set c ⊂ ω thereis a subset c′ ⊂ c such that for every finite set d ⊂ ω, if c ∩

⋂n∈d an /∈ I then

c′ ∩⋂n∈d an /∈ I and if c ∩

⋂n∈d an ∈ I then c′ ∩

⋂n∈d an is finite.

It is not dificult to see that every Fσ ideal is skew and every analytic P-ideal isskew.

Theorem 3.4.9. If I is a skew analytic ideal on ω then I+ is a skew ana-lytic ideal as well. If in addition the equivalence relation =I is placid then theequivalence relation =I+ is placid as well.

Proof. For the first sentence, if c ⊂ ω is a set and c′ ⊂ c witnesses the skew prop-erty for I and c, then c′ also witnesses the skew property for I+ and c. The proofof the second sentence is more involved. Write X = (2ω)ω. Let V [G0], V [G1]


be generic extensions containing respective =I+ -related points x0, x1 ∈ X. As-sume V [G0] ∩ V [G1] = V and work to find a ground model point x ∈ X whichis =I+ -related to both x0, x1.

Consider the set b = {n ∈ ω : x0 � an is I+-equivalent to some element of theground model}. The definition of the set b depends only on the =I+ -equivalenceclass of x0, therefore the set b belongs to both V [G0] and V [G1], and by theinitial assumptions, to V . Let f be the map with domain b which to each n ∈ bidentifies the =I+ -class in V which contains x0 � an. Again, the definition off depends only on the =I+ -class of x0, so f ∈ V [G0] and f ∈ V [G1], thereforef ∈ V . By the Mostowski absoluteness between V and V [G0], there is a pointx ∈ X such that for all n ∈ b, x � an ∈ f(n). We will show that x =I+ x0 holds.

Suppose towards a contradiction that this fails. Consider the set c = {i ∈ω : x(i) 6= x0(i)} /∈ I+ and the set d = {i ∈ ω : x0(i) 6= x1(i)} ∈ I+. It is nowtime to use the skew assumption on the ideal I. Find a set c′ ⊂ c witnessingthe skew property of I+. Use the definition of the jump to find a set e ∈ I suchthat for all n ∈ ω, d ∩ an /∈ I implies e ∩ an 6= 0. Since the set c′ is positive inthe jump, there must be a number n ∈ ω such that c′ ∩ an /∈ I and e ∩ an = 0.The choice of the set c′ shows that c′ ∩ an is in fact I+-positive. The choice ofthe set e shows that d ∩ an ∈ I. This means that x0 � an =I x1 � an, and bythe placid assumption on the equivalence relation =I , n ∈ b must hold. Thisstands in contradiction with the fact that c′ ∩ an, so c ∩ an is I+-positive. Theproof of the theorem is complete.

Example 3.4.10. Let I be the branch ideal on 2ω, and let {an : n ∈ ω} be anenumeration of the basic open subsets of 2<ω. The ideal I is skew. One canstart iterating the jump for countable ordinals α, at limit stages taking unions.The resulting ideals consist of subsets of ω<ω whose closure in 2ω is countablewith Cantor–Bendixson rank ≤ α. All the resulting equivalence relations areplacid.

3.5 Absoluteness

The placid and virtually placid classes of equivalence relations are defined insuch a way that it is not clear whether the membership in them is absolute, andwhat the actual complexity is. This section provides a satisfactory resolutionto these questions.

To reach the absoluteness result, one has to perform a computation of inter-sections of forcing extensions of independent interest. The computation startswith several definitions:

Definition 3.5.1. Let B be a Boolean algebra. A subalgebra A ⊂ B is projec-tive if the projection function π : B → A, assigning to each b ∈ B the smallestelement of A which is ≥ b, is defined for every b.

A good example of a projection is any complete subalgebra of a completeBoolean algebra. The point of the current definition is that the property of


being projective is first order in A,B, so (unlike completeness) transfers frommodel to model without damage. Note that if A ⊂ B is a projective subalgebra,then every maximal antichain of A is also a maximal antichain of B; therefore,the intersection of a generic filter on B with A will be a generic filter on A. Tosee this, let D ⊂ A be a maximal antichain of A and b ∈ B be a condition. Letd ∈ D be an element of A compatible with π(b). Then π(b) − d ∈ A is strictlysmaller element of A than π(b), so π(b)− d 6≥ b holds. This means that d and bmust be compatible.

Definition 3.5.2. Let B be a Boolean algebra and A0, A1 ⊂ B be projectivesubalgebras with projection functions π0, π1.

1. The projection sequence starting at b ∈ B is the sequence 〈bn : n ∈ ω〉defined by b0 = b, b2n+1 = π0(b2n) and b2n+2 = π1(b2n+1);

2. the pair {A0, A1} is projective if for each b ∈ B, the supremum of theprojection sequence starting at b exists in B and belongs to A0 ∩A1.

Again, a good example of a projective pair is a pair of complete subalgebras ofa complete Boolean algebra. The notion of a projective pair is no longer a firstorder statement about A0, A1, B, but it still transfers between ω-models of ZFCwithout damage.

Proposition 3.5.3. Let B be a Boolean algebra and {A0, A1} be a projectivepair of subalgebras. Then

1. A0 ∩A1 ⊂ B is a projective subalgebra of B;

2. if τ0, τ1 are A0- and A1-names for sets of ordinals and B τ0 = τ1, thenthere is a A0 ∩A1-name τ2 such that B τ0 = τ1 = τ2.

Proof. Let π0, π1 be the projections of B to A0, A1. Let π : B → A0∩A1 be thefunction which assigns to each b ∈ B the supremum of the projection sequencestarting from b. It is clear from the definitions that π(b) is the smallest elementof A0 ∩A1 above b and (1) follows.

For (2), first note that for every b ∈ B and every ordinal α, b α ∈ τ0 just incase π(b) α ∈ τ . To see this, by induction on n ∈ ω argue that bn α ∈ τ0. Ifthis holds for n = 2k, then use the fact that τ0 is an A0-name, so bn+1 = π0(bn)must force α ∈ τ ; if this holds for n = 2k + 1, then use the fact that τ1 isan A1-name forced to be equal to τ0, so bn+1 = π1(bn) must force α ∈ τ . Itfollows that b α ∈ τ just in case π(b) α ∈ τ , so one can let 〈π(b), α〉 ∈ τ2 ifb α ∈ τ0.

The above notions have the slightly unpleasant feature that they do notnecessarily survive localization well, so an additional local definition is needed.

Definition 3.5.4. Let B be a Boolean algebra and {A0, A1} be a pair of subal-gebras. The pair is locally projective if for every nonzero element b ∈ B, in the al-gebraB � b the subalgebrasA0 � b = {b∧a : a ∈ A0} andA1 � b = {b∧a : a ∈ A1}form a projective pair.


The following is the central motivation of the notion of a projective pair ofsubalgebras:

Proposition 3.5.5. Let B be a Boolean algebra and A0, A1 be a locally projec-tive pair of subalgebras. The following are equivalent:

1. B V [G ∩A0] ∩ V [G ∩A1] = V ;

2. for every b ∈ B, the intersection algebra A0 � b ∩A1 � b has an atom.

Proof. Suppose that (2) fails for some b ∈ B. Write C = A0 � b ∩ A1 � b anduse Proposition 3.5.3 to argue that C is a projective subalgebra of B � b, sob G ∩ C is a C-generic filter, and since C is nonatomic, G ∩ C /∈ V . At thesame time, G ∩ C belongs to V [G ∩A0] ∩ V [G ∩A1], so (1) fails.

Suppose now that (2) holds. Suppose that τ0, τ1 are A0- and A1-names forsets of ordinals and some condition b ∈ B forces τ0 = τ1. Write C = A0 �b ∩ A1 � b and let c be an atom of C. To verify (1), it is enough to arguethat c decides the membership of any given ordinal in τ0. However, this followsimmediately from Proposition 3.5.3 (2) applied below b.

Finally, we are in a position to give a succinct and principled proof of the mainabsoluteness result of this section.

Theorem 3.5.6. Let E be a Borel equivalence relation on a Polish space X.The statement “E is virtually placid” is absolute among transitive models of settheory containing the code for E.

Proof. We will show that the statement “E is not virtually placid” is in ZFCequivalent to the statement “there is a countable ω-model M of a large fragmentof ZFC containing the code for E such that M |= E is not virtually placid”.This is an analytic statement; therefore, it is absolute among transitive modelsof set theory.

Now, the left-to-right implication is immediate: one just needs to take acountable elementary submodel of a large enough structure to get the requi-site M . The right-to-left direction is more interesting. Suppose that M is acountable ω-model containing the code for E which satisfies that E is not vir-tually placid. Working in the model M , there must be complete algebra B,complete subalgebras A0, A1 ⊂ B, and respective A0, A1-names τ0, τ1 for ele-ments of the underlying space X such that first, B × B ¬(τ0)left E (τ0)right,

and second, B V [G ∩ A0] ∩ V [G ∩ A1] = V and τ0 E τ1. The subalgebrasA0, A1 ⊂ B form a locally projective pair in M since they are complete. ByProposition 3.5.5 applied in M in the (1)→(2) direction, for every b ∈ B, thealgebra A0 � b ∩ A1 � b has an atom. Stepping out of the model M , we seethat the pair A0, A1 ⊂ B is a projective pair and for every b ∈ B, the algebraA0 � b ∩ A1 � b has an atom. By Proposition 3.5.5 applied in the (2)→(1)direction, B V [G ∩ A0] ∩ V [G ∩ A1] = V holds. Moreover, B τ0 E τ1, andB ×B ¬(τ0)left E (τ0)right, since E is absolute between generic extensions ofV and generic extensions of M by Borel absoluteness. In conclusion, E is notvirtually placid in V .

3.6. A VARIATION FOR MEASURE 97

Corollary 3.5.7. Let E be a Borel equivalence relation on a Polish space X.The statement “E is placid” is absolute among transitive models of set theorycontaining the code for E.

Proof. The placidity of E is a conjunction of virtual placidity and the pinnedproperty of E by Proposition 3.3.4. The conjuncts are absolute by Theorem 3.5.6and 2.7.1, and so is the conjunction.

3.6 A variation for measure

The purpose of this section is to introduce a parallel for turbulence for measure,with attendant ergodicity results for measure. Curiously enough, the measuretheoretic parallel for turbulence is intimately connected to the concentration ofmeasure phenomenon. The following definition is close to the whirly actions of[33]:

Definition 3.6.1. Let X be a Polish space with a Borel probability measure µand a metric d. Let Γ be a Polish group acting on X in a measure preservingand distance preserving fashion. We say that the action has concentration ofmeasure if for every open neighborhood U ⊂ Γ containing the unit and everyε > 0 there is δ > 0 such that for every d-ball B ⊂ X of radius < δ and everyBorel set C ⊂ B of relative µ-mass > ε, the set (U ·C) ∩B has relative µ-mass> 1/2.

To formulate the main results of this section, let PΓ be the Cohen forcing onΓ and Pµ be the random forcing with µ, i.e. the forcing with µ-positive Borelsubset of X, ordered by inclusion. The poset PΓ adds a generic point γgen ∈ Γwhile the poset Pµ adds a random point xgen ∈ X.

Theorem 3.6.2. Suppose that Γ is a Polish group acting on a Polish space Xwith a Borel probability measure µ and an ultrametric d in a measure preservingand distance preserving fashion. Suppose that the action has concentration ofmeasure. Then PΓ × Pµ V [xgen ] ∩ V [γgen · xgen ] = V .

The proof of Theorem 3.6.2 hinges on a new walk concept and a proposition.

Definition 3.6.3. Let U ⊂ Γ be an open neighborhood of the unit and δ > 0.A U, δ-walk is a sequence 〈xi : i ≤ j〉 of points in X such that for every i ∈ j,either d(xi, xi+1) < δ or there is γ ∈ U such that γ · xi = xi+1.

Definition 3.6.4. Let U ⊂ Γ be an open neighborhood of the unit and D ⊂ Xbe a closed set. We say that the set D is U -connected if for any two pointsx0, x1 ∈ D and any δ > 0 there is a U, δ-walk from x0 to x1 using only pointsfrom D.

The main import of the concentration of measure assumption is that for anyopen neighborhood U of the unit, closed U -connected µ-positive sets can befound under every stone.


Proposition 3.6.5. Suppose that the action of Γ has concentration of measureand d is an ultrametric. Then for every open neighborhood U ⊂ Γ of the unitand every µ-positive Borel set C ⊂ X there is a µ-positive U -connected compactset D ⊂ C.

Proof. We will first fix useful terminology. For a symmetric neighborhood V ⊂ Γcontaining the unit and d-balls B0, B1 ⊂ X of the same radius we say that B0

is V -related to B1 if there is an element γ ∈ V such that γ ·B0 = B1. Since d isan ultrametric, this is equivalent to the statement that there is γ ∈ V such thatγ · B0 ∩ B1 6= 0. A set B consisting of d-balls of the same radius will be calledV -connected if for any two balls B0, B1 ∈ B one can find a sequence of ballsin B which starts with B0, ends in B1, and successive balls in it are V -related.Lastly, for Borel sets B,D ⊂ X write µB(D) for the relative measure of D in

B, i.e. the ratio µ(B∩D)µ(B) .

Let U ⊂ Γ be an open neighborhood of the unit, and let C ⊂ X be a Borelµ-positive set. Find a symmetric open neighborhood V ⊂ Γ of the unit suchthat V 4 ⊂ U , and thin down C if necessary to a compact µ-positive set. Letε0 = 1/8 and for each n ∈ ω let εn+1 = 2−n−2εn. Let δn > 0 be the numberswitnessing the concentration of measure of the action for V and εn. Thinningdown C if necessary, use the Lebesgue density theorem to find a d-ball Bini ofradius < δ0 such that C ⊂ Bini and µBini

(C) > 1/2.By recursion on n ∈ ω find finite families Bn of pairwise disjoint d-balls of

radius < δn such that for every number n ∈ ω Bn+1 refines Bn, and writingY (B) = {A ∈ Bn+1 : A ⊂ B for every ball B ∈ Bn, the following hold:

• for every A ∈ Y (B), µA(C) > 2n+2εn+1;

• µB(C ∩⋃Y (B)) > (1− 2−n)µB(C).

• the set Y (B) consists of balls of the same radius < δn+1 and it is V 2-connected.

To begin, set B0 = {Bini}. Now, suppose that Bn has been constructed andB ∈ Bn is a ball; we shall show how to construct the set Y (B) and thereforeBn+1. First, use the compactness of the set C to find a finite set Y0 of d-balls (subsets of B) of the same radius < δn+1 such that C ∩ B ⊂

⋃Y0. Let

Y1 = {A ∈ Y0 : µA(C) > 2n+2εn+1}. Thus, µB(C∩⋃

(Y0\Y1)) ≤ 2n+2εn+1 = εn.

Claim 3.6.6. There is a V 2-connected component Y2 ⊂ Y1 such that µB(C ∩⋃(Y1 \ Y2)) < εn.

Proof. We first show that there is a V 2-connected component Y2 ⊂ Y1 such thatµB(C ∩

⋃Y2) > εn. If this were not the case, it would be possible to divide

Y1 into sets Y ′ and Y ′′ which are both invariant under V 2-connections, andµB(C∩

⋃Y ′) and µB(C∩

⋃Y ′′) are both greater than εn. By the concentration

of measure assumption, the sets V · (C ∩⋃Y ′) and V · (C ∩

⋃Y ′′) are both

of µB-mass greater than 1/2 and therefore intersect. It follows that V 2 · (C ∩


⋃Y ′) ∩ (C ∩

⋃Y ′′) 6= 0, contradicting the invariance of the sets Y ′ and Y ′′

under V 2-connections.

Now, the V 2-connected component Y2 found in the first paragraph must infact be such that µB(C ∩

⋃(Y1 \Y2)) < εn, by an argument identical to the one

in the previous paragraph with Y ′ = Y2 and Y ′′ = Y1 \ Y2. This completes theproof of the claim.

Letting Y (B) = Y2 completes the construction. The V 2-connectedness is clearfrom the choice of Y2, the first item is clear from the choice of Y1(B), andthe second item follows from some arithmetic: µB(C ∩

⋃(Y \ Y2)) < 2εn =

2−n2n+1εn ≤ 2−nµB(C).

Now, let D =⋂n

⋃Bn. The set D is closed as the d-balls in the sets Bn are

clopen. It is also a subset of C, because each of the balls in Bn has nonemptyintersection with C, and C is closed. The set D also has positive µ-mass; themass distribution of D is governed by the following claim:

Claim 3.6.7. For every n ≥ 2 and every ball B ∈ Bn, µB(D) > εn.

Proof. To get the set D ∩ B, we subtracted from C ∩ B a set of size at mostΣm>n2−mµB(C) by the second item above, so µB(D) > 1/2µB(C) > εn by thefirst item above.

Towards the connectedness of the set D, consider the following claim:

Claim 3.6.8. Let m ≤ n be natural numbers, B ∈ Bm a ball, and x0, x1 ∈ B∩Dbe any points. Then there is a U, δn-walk from x0 to x1 using only points inB ∩D.

Proof. This is proved by induction on n −m, which is the reason for the con-voluted statement of the claim. The case n − m = 0 is trivial since thend(x0, x1) < δn. Now suppose that the statement is known for m + 1 ≤ nand proceed to show that it is also true for m ≤ n. Let B0, B1 ∈ Bm+1 beballs such that x0 ∈ B0 and x1 ∈ B1. If B0 = B1 then the induction hypoth-esis immediately applies. Otherwise, by the third item above, the set Y (B) isV 2-connected, so there must be a sequence of balls in Y (B) starting at B0 andending at B1 such that successive balls are V 2-connected. Suppose for simplic-ity that this sequence has length 2–i.e. B0 and B1 are V 2-connected. Fix anelement γ ∈ V 2 such that γ ·B0 = B1.

Since µB0(D) and µB1

(D) are both greater than εm+1, the concentrationassumption yields that the numbers µB0

(V · (D ∩B0)) = µB1(V · (D ∩B1)) are

both greater than 1/2. It follows that there is a point x ∈ (V · (D ∩ B0)) ∩ B0

such that γ · x ∈ (V · (D ∩B1)). Find points x′0 ∈ D ∩B0 and x′1 ∈ D ∩B1 andgroup elements β0, β1 ∈ V such that β0 · x′0 = x and β1 · (γ · x) = x′1. Use theinduction hypothesis to find a U, δn-walk from x0 to x′0. Follow it by acting byβ0γβ1 ∈ U to get to the point x′1, and then use the induction hypothesis againto extend the walk from x′1 to x1. The claim follows.


Since the numbers δn tend to 0 as n tends to infinity and D ⊂ Bini, theclaim shows that the set D is U -connected and completes the proof of theproposition.

Proof of Theorem 3.6.2. Suppose towards a contradiction that 〈U,C〉 is a con-dition in the product PΓ × Pµ which forces V [xgen ] ∩ V [γgen · xgen ] 6= V . Thefollowing claim follows from abstract forcing considerations.

Claim 3.6.9. 〈U,C〉 2ω ∩ V [xgen ] ∩ V [γgen · xgen ] 6= 2ω ∩ V .

Proof. It will be enough to show that Pµ forces that for every ordinal α andevery function f : α→ 2 in the extension, if f /∈ V then there is a ground modelcountable set b ⊂ α such that f � b /∈ V . It will follow immediately that PΓ×Pµforces that if f ∈ V [xgen ]∩V [γgen ·xgen ]\V then f � b ∈ V [xgen ]∩V [γgen ·xgen ]\Vfor some ground model countable set b. This will prove the claim.

The forcing property of Pµ in question is well-known; we include a completeproof. Suppose towards a contradiction that B ∈ Pµ and B τ : α → 2is a function which is not in the ground model, and for every countable setb ∈ V , τ � b ∈ V . Let 〈Mβ : β ∈ ω1〉 be an ∈-tower of countable elementarysubmodels of a large structure containing B, τ as elements. For each β ∈ ω1 usethe contradictory assumption to find a function gβ : Mβ ∩ α → 2 in the modelMβ+1 such that some condition below B forces τ � Mβ = gβ . Let Bβ ⊂ B bethe Borel set representing the nonzero Boolean value of the latter statement;Bβ ∈Mβ+1 holds by elementarity of the model Mβ+1, but also Bβ /∈Mβ sinceτ is forced not to belong to the ground model. Use a counting argument andthe Lebesgue density theorem to find a basic open set O ⊂ X such that theset C = {β ∈ ω1 : µ(Bβ ∩ O) > 1

2µ(O)} is uncountable. Since the conditions{Bβ : β ∈ C} are pairwise compatible, the functions {gβ : β ∈ C} must form anincreasing chain, so in fact the conditions {Bβ : β ∈ C} form a strictly decreasingchain in Pµ. This contradicts the countable chain condition of Pµ.

Strengthening the condition 〈U,C〉 if necessary, we may find a continuous func-tion f : C → 2ω and a name τ in the complete Boolean algebra generated by thename for γgen · xgen such that the fibers of f are µ-null and 〈U,C〉 f(xgen) =τ/γgen · xgen . Let M be a countable elementary submodel of a large structurecontaining f, τ in particular, and let C ′ ⊂ C be a set of points Pµ-generic overthe model M . The set C ′ is Borel and µ-positive. Find open subsets V,U ′ ofΓ such that 1 ∈ V , U ′ ⊂ U , and U ′ · V ⊂ U . Use Proposition 3.6.5 to find aV -connected compact µ-positive subset D ⊂ C ′.

Since the fibers of f are µ-null, there must be points x0, x1 ∈ D such thatf(x0) 6= f(x1). Find a number n ∈ ω such that f(x0)(n) 6= f(x1)(n). LetO0 = {x ∈ D : f(x)(n) = f(x0)(n)} and O1 = {x ∈ D : f(x)(n) = f(x1)(n)}.These are complementary relatively open subsets of the compact set D, so theyare separated by some distance δ > 0. Use the connectedness of the set D toproduce a V, δ-walk from x0 to x1. There must be successive points x′0, x′1 onthe walk such that f(x′0)(n) 6= f(x′1)(n). The two points are at a distance > δby the choice of δ, so there must be a group element β ∈ V such that β ·x′0 = x′1.


By the Baire category theorem, there must be an element γ ∈ U ′ whichbelongs to no meager subset of Γ coded in the model V [x′1] and also to no rightβ−1-shift of any meager subset of Γ coded in the model V [x′0]. As a result, thepoint γ is PΓ-generic over V [x′1] and the point γ ·β is PΓ-generic over the modelV [x′0]. Both of these points belong to the set U . By the product forcing theorem,the pairs 〈γβ, x′0〉 and 〈γ, x′1〉 are both PΓ × Pµ-generic over the model M ,meeting the condition 〈U,C〉. However, f(x′0) 6= f(x′1) while τ/γβx′0 = τ/γx′1,violating the forcing theorem in view of the initial contradictory assumption.Theorem 3.6.2 follows!

The main corollaries are encapsulated in the following ergodicity result.

Corollary 3.6.10. Suppose that Γ is a Polish group acting on a Polish space Xwith a Borel probability measure µ and an ultrametric d in a measure preservingand distance preserving fashion. Suppose that the action has concentration ofmeasure. Suppose that E is the orbit equivalence relation, Y is a Polish space,F on Y is an analytic virtually placid equivalence relation, and h : X → Y is aBorel homomorphism from X to Y . Then there is an F -equivalence class withµ-positive h-preimage.

Proof. Let γ ∈ Γ and x ∈ X be mutually PΓ-generic and Pµ-generic points,and look at the models V [x] and V [γ · x]. Since h is a homomorphism of E toF and x E γ · x, h(x) F h(γ · x) must hold. Since F is virtually placid andV [x]∩ V [γ · x] = V holds per Theorem 3.6.2, there must be a virtual F -class inthe ground model such that h(x) and h(γ ·x) realize it. Since the poset PΓ×Pµis c.c.c., Theorem 2.6.2 shows that all virtual classes realized in its extension arein fact repreented in the ground model. Thus, there is a ground model elementy ∈ Y such that h(x) F y holds. Since x is a Pµ-generic point, it belongs to noanalytic ground model coded µ-small sets. Thus, µ(h−1[y]F ) > 0 as desired.

Examples of actions with concentration of measure are not easy to identify. Thefollowing examples use Fσ P-ideals I on ω (which are Polish groups with thesymmetric difference operation and a suitable topology by a result of Solecki[85]) and their standard action on 2ω (a · x = y just in case {n ∈ ω : x(n) 6=y(n)} = a}), inducing the equivalence relation =I . The action preserves theusual Borel probability measure µ on 2ω and also the usual minimum differencemetric d on 2ω.

Example 3.6.11. Let {an : n ∈ ω} be positive real numbers such that Σnan isinfinite while Σna

2n is finite. Let I be the ideal of all sets b ⊂ ω such that Σn∈ban

is finite. One can view I as a Polish group Γ with the complete metric e(γ, δ) =Σ{an : γ(n) 6= δ(n)}, continuously acting on the spaceX = 2ω by coordinatewiseBoolean addition. The action exhibits the concentration of measure.

To see this, let U be a neighborhood of the unit in Γ, and ε > 0 be a realnumber. Find a real number η > 0 such that the η-ball in the metric e around theunit is a subset of U , and find a number m ∈ ω such that 2exp(−η2/8Σ∞n=ma

2n) <

ε. The concentration of measure formula in [73, Theorem 4.3.19] then showsthat δ = 2−m works as required in Definition 3.6.1.


Corollary 3.6.12. Let I be the usual summable ideal on ω. Let h : 2ω → X bea Borel homomorphism of =I to a virtually placid analytic equivalence relationF on a Polish space X. Then there is an F -class whose h-preimage has fullµ-mass.

Proof. By Corollary 3.6.10, there is an F -class whose h-preimage has positive µ-mass. However, the h-preimage is an =I -invariant set, the equivalence relation=I includes E0 as a subset, and by the usual E0-ergodicity considerations, theh-preimage must in fact have full µ-mass.

The concentration of measure for actions fails in many cases. Typically, there isa homomorphism of the orbit equivalence relation which violates the conclusionof Corollary 3.6.10 and therefore witnesses the failure of the concentration ofmeasure in a strong sense.

Example 3.6.13. There is a summable-type ideal I on ω and a Borel homo-morphism h : 2ω → 2ω of =I to E0 such that preimages of E0-equivalence classesare µ-null.

Proof. The key tool is the following:

Claim 3.6.14. For every i ∈ ω and every ε > 0 there is a number n ∈ ω andsets a, b ⊂ 2n of the same relative size > 1−ε

2 each, such that for every x ∈ aand y ∈ b the set {m ∈ n : x(m) 6= y(m)} contains at least i many elements.

Proof. Fix i and ε. Stirling’s approximation formula shows that there is n ∈ ωsuch that the size of the set {a ⊂ n : ||a| − n

2 | < i + 1} is less than ε2n. Leta = {x ∈ 2n : the set {m ∈ n : x(m) = 1} contains at most n

2 −i many elements}and b = {x ∈ 2n : the set {m ∈ n : x(m) = 1} contains at least n

2 + 1 manyelements}. This works.

Towards the proof of the example, find a partition ω =⋃n In into finite sets

such that for every n ∈ ω, the set 2In contains subsets an, bn of the same sizesuch that their relative size in 2In is greater than 1/2− 2−n, and if x ∈ an andy ∈ bn are arbitrary elements, then the set {i ∈ In : x(n) 6= y(n)} has size atleast n. Now, define w(m) = 1

n+1 if m ∈ In and I = {a ⊂ ω : Σn∈aw(n) <∞}.Define B = {x ∈ 2ω : ∀∞n x � n ∈ an ∪ bn}; this is a Borel set of full mass. Forx ∈ B, define h0(x) ∈ 2ω by h0(x)(n) = 0 ↔ x � In ∈ an. It is not difficult tocheck that h0 : B → 2ω is a continuous homomorphism from B to E0 such thatpreimages of E0-classes are of zero mass. The rest of the proof only adjusts h0

to a Borel homomorphism h defined on the whole space.To this end, let Cn for n ∈ ω be inclusion increasing compact subsets of B

whose mass converges to 1. The set⋃n Cn is Kσ and the equivalence relation

=I is Kσ, so the saturation D = [⋃n Cn]=I is Kσ as well. Let F ⊂ D×2ω be the

Borel set of all pairs 〈x, y〉 such that for some (equivalently, for all) x′ ∈ B suchthat x′ =I x, h0(x′) is E0-related to y. F is a Borel set with nonempty countablesections, and by the Lusin–Novikov theorem, it has a Borel uniformization h.Extend h to all of 2ω by defining h(x) for x /∈ D to be an arbitrary fixed elementof 2ω. It is not difficult to check that h has the required properties.


Example 3.6.15. There is a Tsirelson-type [27] ideal I on ω and a Borelhomomorphism h : B → 2ω of =I to E0 such that preimages of E0-equivalenceclasses are µ-null.

Proof. We will deal with a certain special kind of Tsirelson submeasures. Letα > 0 be a real number and f : ω → R+ be a nonincreasing function con-verging to 0. By induction on n ∈ ω define submeasures νn on ω by settingν0(a) = supi∈a f(i), and νn+1(a) = sup{νn(a), α

∑b∈~b νn(b)} where the variable

~b ranges over all finite sequences 〈b0, b1, . . . bj〉 of finite subsets of a such thatj < min(b0) ≤ max(b0) < min(b1) ≤ max(b1) < . . . . In the end, let the sub-measure ν be the supremum of νn for n ∈ ω. Some computations are necessaryto verify that ν is really a lower semicontinuous submeasure on ω [27]. TheTsirelson ideal I = {a ⊂ ω : limm ν(a \m) = 0} turns out to be an Fσ P-ideal[27].

By induction on i ∈ ω choose intervals Ii ⊂ ω such that max(Ii) < min(Ii+1)and such that min(Ii) > i/α and there are sets ai, bi ⊂ 2Ii of the same relative

size ≥ 1−2−i

2 such that for any elements x ∈ ai, y ∈ bi the set {m ∈ Ii : x(m) 6=y(m)} has size at least i/α. This is easily possible by Claim 3.6.14. Now,consider the function f defined by f(m) = 1/i for m ∈ (max(Ii−1),max(Ii)]and let ν be the derived submeasure and I the derived Tsirelson ideal. Observethat with this choice of the function f , for any i ∈ ω and elements x ∈ ai, y ∈ bithe set {m ∈ Ii : x(m) 6= y(m)} has ν-mass at least 1, since it has ν1-mass atleast 1. The rest of the proof follows the lines of Example 3.6.13.

Example 3.6.16. Let J be the Rado graph ideal on ω. There is a Borelhomomorphism h : 2ω → 2ω from =J to E0 such that preimages of E0-classesare µ-null.

Proof. Let G be the Rado graph, interpreted so that ω is the set of its vertices;then J is the ideal on ω generated by J-cliques and J-anticliques. To constructh, by induction on n ∈ ω find pairwise disjoint finite sets In ⊂ ω and setsan, bn ⊂ 2In such that

• each In is a G-anticlique;

• if n 6= m then In × Im ⊂ G;

• for every x ∈ an and every y ∈ bn the set {t ∈ In : x(t) 6= y(t)} has size atleast n;

• the sets an, bn ⊂ 2In have the same relative size larger than 1/2− 2−n.

This is easy to do using the universality properties of the Rado graph andClaim 3.6.14 repeatedly. Let B = {x ∈ 2ω : ∀∞n x � In ∈ an ∪ bn} andlet h0 : B → 2ω be the Borel map defined by h0(x) = 0 ↔ x � In ∈ an. It isimmediate that the set B has full µ-mass and the function h is a homomorphismfrom =J to E0. The rest of the proof follows the lines of Example 3.6.13.


There are numerous questions left open by the development of this section, ofwhich we quote two.

Question 3.6.17. Can the assumption that d be an ultrametric be eliminatedfrom the assumptions of Theorem 3.6.2?

Question 3.6.18. Is there a Tsirelson ideal whose natural action on 2ω exhibitsconcentration of measure?

Chapter 4

Nested sequences of models

4.1 Prologue

The purpose of this chapter is to set up a calculus for infinite nested sequencesof models of ZFC, which turn out to be critical for the treatment of the E1

cardinal. As a motivation, we include a simple proof of the fact that E1 is notBorel reducible to any orbit equivalence relation. It is quite different from thestandard one [48, Theorem 11.8.1], and it has the advantage that it can be easilyadapted to the context of inequalities between cardinalities of quotient spaces.

Theorem 4.1.1. E1 is not Borel reducible to any orbit equivalence relation.

Proof. Let Y be the Polish space (2ω)ω and Γ be a Polish group continuouslyacting on a Polish space Z, inducing the orbit equivalence relation F . Supposetowards a contradiction that there is a Borel reduction h : Y → Z of E1 to F .Let 〈xn : n ∈ ω〉 be a sequence of mutually Cohen generic points in 2ω. For eachn ∈ ω let yn denote the element of Y for which y(m) is the zero sequence in 2ω

if m < n, and xm if m ≥ n; write Mn = V [yn].Work in the model M0. The reinterpretation of the Borel map h in M0 is

still a reduction of E1 to F . For each number n > 0 fix a group element γn ∈ Γsuch that γn ·h(yn) = h(y0). Let γ ∈ Γ be a point Cohen-generic over V [y0] andlook into the model V [γ · h(y0)]. By a Mostowski absoluteness argument, theremust be a point y ∈ V [γ · h(y0)] such that h(y) F γ · h(y0). Since the functionh is a reduction of E1 to F even in the model M0[γ], this point y ∈ Y mustbe E1-related to y0, so for all but finitely many numbers n it must be the casethat y(n) = xn. Choose a number n ∈ ω such that y(n) = xn and look at themodel Mn+1. The point γγn+1 ∈ Γ is Cohen generic over M0 by the invarianceof the meager ideal under right shift, and by the product forcing theorem itfollows that the models M0 and Mn+1[γγn+1] are mutually generic over themodel Mn+1. Now, the points yn+1 and γγn+1 · h(yn+1) = γ · h(y0) belong tothe model Mn+1[γγn+1] and so does y. Thus, even the point y(n) = xn ∈ 2ω

belongs to this model; however, it is a point of M0 Cohen generic over Mn+1.This contradicts the product forcing theorem.

105

106 CHAPTER 4. NESTED SEQUENCES OF MODELS

4.2 Coherent sequences of models

It is clear from the proof of Theorem 4.1.1 that its generalizations will requirecodification of decreasing ω-sequences of generic extensions. In addition to theapproach from Theorem 4.1.1, we pay close attention to the intersection model.This is the content of the following definitions and theorems.

Definition 4.2.1. Let 〈Mn : n ∈ ω〉 be an inclusion decreasing sequence oftransitive models of ZFC. We say that the sequence is coherent if for everyordinal λ and every natural number n, the sequence 〈Vλ ∩Mm : m ≥ n〉 belongsto Mn. Given a coherent sequence of models 〈Mn : n ∈ ω〉, a sequence 〈vn : n ∈ω〉 is coherent if for every number m ∈ ω, 〈vn : n ≥ m〉 ∈Mm holds.

Example 4.2.2. Let Rm for m ∈ ω be any partial orders and let 〈Gm : m ∈ω〉 be a sequence of generic filters on the respective posets Rm added by thecountable support product

∏mRm. Let Mn = V [Gm : m ≥ n]. Then 〈Mn : n ∈

ω〉 is a coherent sequence of models.

Example 4.2.3. Let x be a set. By recursion on n ∈ ω, define models Mn

by letting M0 be V and each Mn+1 be the collection of all sets hereditarilydefinable from ordinal parameters and the parameter x∩Mn in the model Mn.The sequence 〈Mn : n ∈ ω〉 is coherent.

Example 4.2.4. Let κ be a measurable cardinal, U a measure on it, andj : V → M the U -ultrapower, with iterands denoted by jα for every ordinal α.For each n ∈ ω let Mn = jn(V ). The sequence 〈Mn : n ∈ ω〉 is coherent.

Most of our choice-coherent sequences are sequences of generic extensions in thefollowing sense:

Definition 4.2.5. A sequence 〈Mn : n ∈ ω〉 is generic over V if V is a modelof ZFC contained in all Mn for n ∈ ω and M0 is a generic extension of M .

The usual abstract forcing arguments (Fact 1.7.6) show that if the sequence ofmodels is generic over V then all models Mn are generic extensions of V and ifn ∈ m are numbers then Mn is a generic extension of Mm. Coherent sequencesof models are most often formed as generic extensions of the constant sequence〈Mn = V : n ∈ ω〉 using the following definitions and theorem.

Definition 4.2.6. Let P,Q be posets. A projection of Q to P is a pair oforder-preserving functions π : Q→ P and ξ : P → Q such that

1. π ◦ ξ is the identity on P ;

2. whenever π(q) ≤ p then q ≤ ξ(p);

3. whenever p ≤ π(q) then there is q′ ≤ q such that π(q′) ≤ p.

As the simplest initial example, let P,R be any posets, let R have a largestelement 1R and let Q = P × R. Then one can consider the projection of Q toP by setting π(p, r) = p and ξ(p) = 〈p, 1R〉. An important effect of the demand(3) is that the π-image of the generic filter on Q is a generic filter on P .

4.2. COHERENT SEQUENCES OF MODELS 107

Definition 4.2.7. Let 〈Mn : n ∈ ω〉 be a coherent sequence of models of ZFC.A coherent sequence of posets is a sequence 〈Pn, πnm, ξmn : n ≤ m ∈ ω〉 suchthat

1. For all numbers n ≤ m the maps πnm : Pn → Pm and ξmn : Pm → Pnform a projection of Pn to Pm;

2. the functions πnm form a commutative system, the same for ξmn, and forall n ∈ ω the functions πnn, ξnn are the identity on Pn;

3. for every number k ∈ ω, the sequence 〈Pn, πnm, ξmn : k ≤ n ≤ m ∈ ω〉belongs to the model Mk.

In particular, every commutative sequence of projections is coherent over theconstant coherent sequence 〈Mn = V : n ∈ ω〉. As the simplest initial example,let 〈Rn : n ∈ ω〉 be a sequence of posets with respective largest elements 1n,let Pn be the finite (or countable) support product of Rm for m ≥ n, andlet πnm(p) = p � [m,ω) and ξnm(p) = 〈1k : k ∈ [n,m)〉ap. The sequence〈Pn, πnm, ξmn : n ≤ m ∈ ω〉 is coherent over 〈Mn = V : n ∈ ω〉.

Theorem 4.2.8. Let 〈Mn : n ∈ ω〉 be a coherent sequence of models of ZFCand 〈Pn, πnm : Pn → Pm, ξmn : Pm → Pn : n ≤ m ∈ ω〉 be a coherent sequenceof posets. Let G ⊂ P0 be a filter generic over M0, and for each n ∈ ω letGn = ξ−1

n0 G. The sequence 〈Mn[Gn] : n ∈ ω〉 is a coherent sequence of modelsof ZFC again.

Proof. Observe that for every number k ∈ ω, in the model Mk[Gk], one canform the sequence 〈Gn : n ≥ k〉 as 〈ξ−1

nkGk : n ≥ k〉 by the commutativity of theξ-maps. Thus, if λ is a limit ordinal larger than the rank of all the posets on thecoherent sequence, one can form the relation {〈n, x〉 : n ≥ k, x ∈ Mn[Gn] hasrank< λ} in the modelMk[Gk] as the set {〈n, τ/Gn〉 : n ≥ k and τ ∈Mn is a Pn-name of rank < λ} by the coherence of the original sequence 〈Mn : n ∈ ω〉.

The critical object for understanding a coherent sequence of models is the in-tersection model Mω =

⋂nMn. In Example 4.2.2, the intersection model is a

model of ZFC, and it has been discussed in [49, Theorem 9.3.4]; we will comeback to it below–Theorem 4.3.5. In the context of general coherent sequences,the intersection model is a transitive model of ZF, and the Axiom of Choicemay fail in it. This is the content of the following theorem and example.

Theorem 4.2.9. If 〈Mn : n ∈ ω〉 is a coherent sequence of generic extensionsof V , then Mω =

⋂nMn is a class in all models Mn, and it is a model of ZF.

Proof. We will only show that Mω is a class in M0; it then follows by the sameargument that Mω is a class in each Mn for n ∈ ω. To show that Mω is a modelof ZF, by [45, Theorem 13. 9], it is enough to show that Mω is closed under theGodel operations and it is universal: for every set a ⊂Mω there is a set b ∈Mω

such that a ⊂ b. The closure under the Godel operations follows from the factthat these operations are evaluated in the same way in each model Mn. For the


universality, suppose that a ⊂ Mω is a set, choose an ordinal α such that allsets in a have rank smaller than α, and form the set b = Vα ∩Mω. Since Mω

is a class in each model Mn, the set b is in all models Mn and therefore in Mω.Clearly, a ⊂ b, concluding the proof of universality.

To show that Mω is a class in M0, let λ be a large limit cardinal in V sothat M0 is a generic extension of V by a poset in Vλ, and such that Vλ satisfiesa large fragment of ZFC. Note that then, all the models Mn are also obtainedfrom V as generic extensions by posets in Vλ. Move to the model M0. Let f bethe function from ω to M0 ∩Vλ such that f(n) = {〈P,G〉 : P ∈ V ∩Vλ, G ∈Mn

is a filter on P generic over V , and whenever 〈Q,H〉 is a pair consisting of aposet in V ∩ Vλ and a filter on Q in Mn generic over V , there is a P -name σin V such that σ/G = H}. The coherence of the sequence 〈Mn : n ∈ ω〉 showsthat the function f can be formed in M0. Then, Mω is exactly the collectionof all sets x such that for every n ∈ ω and every pair 〈P,G〉 ∈ f(n), x ∈ V [G].Since V is a class in M0, this shows that Mω is a class in M0 as well.

Example 4.2.10. Let κ be a measurable cardinal, with a normal measureU on κ and the associated ultrapower j : V → M . Let jnm : Mn → Mm

be the iterands of j for n ≤ m ≤ ω. Then⋂n∈ωMn = Mω[c] holds where

c = 〈j0n(κ) : n ∈ ω〉 [20, 15]. It is well-known and follows from the geometricdescription of Prikry genericity by Mathias [69] that the set c is generic overthe model Mω for the Prikry forcing associated with the measure j0ωU .

Example 4.2.11. Let c0 : ω1 × ω → 2 be a Cohen-generic map, and let cn =c0 � ω1 × (ω \ n). Let Mn = V [cn]. In the model Mω =

⋂nMn, the chromatic

number of G0 is greater than 2; thus, the Axiom of Choice must fail in Mω.

Proof. Work in V . For each number n ∈ ω, let Pn be the poset of all finitefunctions from ω1× (ω \n) to 2 ordered by extension. Note that for each n ∈ ωthe map cn is Pn-generic over V . For each ordinal α ∈ ω1 and a number n ∈ ω,let dαn be a name for the function defined by letting dαn(m) be 0 if m ∈ n andthe unique value of p(α,m) for all conditions p in the generic filter with the pair(α,m) in their domain if m ≥ n. Note that dαn is really a Pn-name and it isforced to belong to the intersection model Mω.

Now, let p ∈ P be a condition and σ be a name such that p σ : 2ω → 2 is afunction in Mω; we will find an ordinal α ∈ ω1, a number n ∈ ω and a conditionstrengthening p which forces dα0 and dαn to differ in an even number of entriesif and only if σ(dα0) 6= σ(dαn). This cannot occur if σ is a coloring of G0.

By a standard ∆-system argument, strengthening p if necessary, we mayfind an infinite set S ⊂ ω1, conditions pα ∈ P for α ∈ S and a number n ∈ ωsuch that the conditions pα for α ∈ S form a ∆-system with root p, dom(pα) ⊂ω1 × n − 1, and each pα decides the value of σ(dα0) to be some bit bα ∈ 2.Find a condition q ≤ p and a Pn-name τ such that q σ = τ ; this is possibleas σ is forced to belong to Mω. Since the set S is infinite, it is possible tofind an ordinal α ∈ S such that pα is compatible with q. Find a conditionr ∈ Pn such that r ≤ q � ω1 × (ω \ n) and r decides the value of τ(dαn) to besome specific bit b ∈ 2. Note that pα and r are compatible in P , and the pair

4.3. CHOICE-COHERENT SEQUENCES OF MODELS 109

〈α, n− 1〉 does not belong to dom(pα ∪ r). Thus, it is possible to strengthen thecondition pα ∪ r to some s ∈ P such that {α} × n ⊂ dom(s), and cardinality ofthe set {m ∈ n : s(α,m) = 1} is even if and only if bα 6= b. This completes theproof.

4.3 Choice-coherent sequences of models

In most of our examples, we will want to look at sequences of models which havea greater degree of coherence. Certain constructions arising from the axiom ofchoice will have to be performed in a coherent way. The following definitionrecords the demands:

Definition 4.3.1. Let 〈Mn : n ∈ ω〉 be an inclusion decreasing sequence oftransitive models of ZFC. We say that the sequence is choice-coherent if it iscoherent and for every ordinal λ there is a well-ordering ≤λ of Vλ ∩M0 suchthat its intersection with each model Mn belongs to Mn.

In the common case of generic coherent sequences, the choice-coherence can bedetected from the theory of the intersection model as follows:

Theorem 4.3.2. Suppsoe that 〈Mn : n ∈ ω〉 is a generic coherent sequence ofgeneric extensions of V . The following are equivalent:

1. 〈Mn : n ∈ ω〉 is choice-coherent;

2. Mω =⋂nMn is a model of ZFC.

Proof. For the (1)→(2) direction, assume the choice coherence. Let λ be anyordinal. In view of Theorem 4.2.9, we only need to produce a well-ordering≤∗ of Vλ ∩Mω such that ≤∈ Mω. Fix a wellordering ≤ witnessing the factthat 〈Mn : n ∈ ω〉 is a choice-coherent decreasing sequence, and note that≤∗=≤ ∩Mω works as desired. The genericity assumption is not needed forthis direction.

For the (2)→(1) direction, that Mω is a model of ZFC. Let λ be any ordinal.Let κ > λ be a cardinal such that each model Mn is a generic extension of V bya poset of cardinality smaller than λ. Let ≺ be a well-ordering of Vκ in V . Byrecursion on n ∈ ω build a sequence 〈Pn, Gn, τn : n ∈ ω〉 such that Pn is ≺-leastposet in V ∩ Vκ such that Mn is Pn-generic extension of V , Gn ⊂ Pn is a filtergeneric over V such that Mn = V [Gn], τn is the ≺-least Pn-name in V suchthat τn/Gn ⊂ Pn+1 is a filter generic over V such that Mn+1 = V [τn/Gn], andGn+1 = τn/Gn. It is not difficult to see that the tail 〈Pn, Gn, τn : n ≥ m〉 of thesequence can be recovered from Gm, and therefore the sequence is coherent.

Now, let ≤λ be the following well-ordering of Vλ∩M0: first come the elementsof Vλ ∩ Mω, then the elements of Vλ ∩ M0 \ M1, and then the elements ofVλ ∩Mn \Mn+1 in turn. The elements of Vλ ∩Mω are ordered by some well-ordering in Mω which is available as Mω is a model of ZFC. The elements ofMn \Mn+1 are ordered by σn/Gn where σn is the ≺-least name in V ∩ Vκ such


that σn/Gn is a well-ordering of Vλ ∩Mn. It is not difficult to see that ≤λ is awell-ordering of Vλ ∩M0 and ≤λ ∩Mn ∈Mn holds for all n ∈ ω.

Most examples of choice-coherent sequences are generic and obtained from thetrivial one 〈Mn = V : n ∈ ω〉 by a coherent forcing which satisfies a certaindegree of completeness.

Definition 4.3.3. Let 〈Pn, πnm, ξmn : n ≤ m ∈ ω〉 be a commutative system ofprojections from posets Pn to Pm for n ≤ m.

1. The diagonal game is the following infinite game between Players I andII, in round n Player I plays pn ∈ Pn and Player II responds by qn ≤ pn.Additionally, pn+1 ≤ πnn+1(qn). In the end, Player II wins if there is acondition r ∈ P0 such that π0n(r) ≤ qn holds for all n ∈ ω.

2. The sequence is diagonally distributive if Player I has no winning strategyin the diagonal game.

Example 4.3.4. Suppose that 〈Qm : m ∈ ω〉 are arbitrary posets, and letPn =

∏m≥nQm be the countable support product with the natural projection

maps from Pn to Pm for n ≤ m. Player II has a simple winning strategy in thediagonal game in this setup: set qn = pn.

Theorem 4.3.5. Let 〈Mn : n ∈ ω〉 be a choice-coherent sequence of models ofZFC. Let 〈Pn, πnm, ξmn : n ≤ m ∈ ω〉 be a coherent sequence of posets which isdiagonally distributive in M0. Let G ⊂ P0 be a filter generic over M0, and letGn = ξ−1

n0 G. Then

1. the sequence 〈Mn[Gn] : n ∈ ω〉 is choice-coherent;

2. the models⋂nMn and

⋂nMn[Gn] contain the same ω-sequences of ordi-

nals.

Proof. Write Mω =⋂nMn. We start with (1). The main task is to find a

poset Pω ∈ Mω and a filter Gω ⊂ Pω generic over Mω such that⋂nMn[Gn] =

Mω[Gω].Using the choice coherence of the original sequence 〈Mn : n ∈ ω〉, we may

assume that there is a sequence 〈αn : n ∈ ω〉 such that the underlying setof each poset Pn is exactly αn. For each condition p ∈ P0, the ordinal ω-sequence 〈π0n(p) : n ∈ ω〉 belongs to Mω, since for each number k ∈ ω, thetail 〈π0n(p) : n ≥ k〉 is reconstructed as 〈πkn(π0k(p)) : n ≥ k〉 in the model Mk.Similarly, the set Pω = {q ∈

∏n Pn : ∃k ∈ ω ∀n ≥ k q(n) = πkn(q(k))} belongs

to the model Mω. For elements q0, q1 ∈ Pω let q1 ≤ q0 if for all but finitelymany numbers n ∈ ω, q1(n) ≤ q0(n) in the poset Pn, and conclude that theposet 〈Pω,≤〉 belongs to the model Mω.

Define a function π0ω : P0 → Pω by π0ω(p) = q where q(n) = π0n(p), and afunction ξω0 : Pω → P0 by ξω0(q) = ξk0(q(k)) where k ∈ ω is such that for alln ≥ k, πkn(q(k)) = q(n). One can also similarly define maps πnω : Pn → Pω


and ξωn : Pω → Pn. It is a matter of trivial diagram chasing to show that themaps form a commuting system of projections from Pn to Pω and moreoverπnω, ξωn ∈ Mn. Thus, letting Gω = ξ−1

ω0G0, one can conclude that Gω ⊂ Pωis a filter generic over M0 and therefore over Mω. Also Gω ∈

⋂nMn[Gn]

since Gω can be reconstructed in Mn[Gn] as Gω = ξ−1ωnGn. In conclusion,

Gω ∈⋂nMn[Gn].

Finally, we have to prove that every element of the intersection⋂nMn[Gn]

belongs to Mω[Gω]. This is where the diagonal distributivity of the originalposet sequence is used. Suppose that τ ∈M0 is a P0-name for a set of ordinalsand p ∈ P0 is a condition forcing τ ∈

⋂nMn[Gn]; we must produce a condition

p′ ≤ p and a Pω-name τω ∈Mω such that p′ τ = τω/Gω. Consider a strategyby Player I in the diagonalization game in which he plays pn so that p0 ≤ p, τ0 =τ , and there is a Pn+1-name τn+1 ∈ Mn+1 such that pn Pn τn = τn+1/Gn+1.This is possible by the assumption on the name τ . By the diagonalizationassumption, Player II has a counterplay with conditions qn ≤ pn such thatthere is a condition p′ ≤ p for which π0n(p) ≤ qn for all n ∈ ω. Let τω be thePω � π0ω(p′)-name defined by q α ∈ τω just in case ξω0(q) P0

α ∈ τ . Thename τω can be reconstructed in every model Mn by the definition q α ∈ τωjust in case ξωn(q) Pn α ∈ τn by the choice of the strategy for Player I in thediagonalization game. As a result, τω ∈Mω. It is immediate from the definitionof τω that p′ τ = τω/Gω as desired.

Now we are ready to construct the requisite well-orderings verifying thechoice-coherence of the models 〈Mn[Gn] : n ∈ ω〉. Let λ be an ordinal largerthan the ranks of all the posets Pn for n ∈ ω. Let ≤ be a coherent well-ordering of Vλ ∩M0. We will now describe a coherent well-ordering ≤′ of setsof rank < λ in the model M0[G0]. In this well-ordering, the sets in Mω[Gω]come first, ordered by some well-ordering in the model Mω[Gω]. The sets inM0[G0] \M1[G1] come next, well-ordered by their ≤-first P0-name in the modelM0 representing them. The sets in M1[G1] \M2[G2] come next with a similarwell-order, and so on. The coherence of the resulting well-ordering ≤′ is due tothe fact that for each k ∈ ω, the sequence 〈Gn : n ≥ k〉 belongs to the modelMk[Gk].

(2) is much easier. Suppose that τ ∈ M0 is a P0-name for an ω-sequenceof ordinals in the model Mω[Gω] and p ∈ P0 is a condition; we must find acondition r ≤ p and an ω-sequence z ∈ Mω such that r τ = z. Consider astrategy for Player I in the diagonal game in which he plays conditions pn ∈ Pnand on the side produces Pn-names τn ∈ Mn so that p0 ≤ p, τ0 = τ andpn Pn τn = τn+1 evaluated by the πnn+1-image of the generic filter on Pn, andalso pn decides the value τn(n) to be some ordinal z(n). The assumptions onthe name τ shows that this is a valid strategy. The initial assumptions on thecoherent sequence of posets show that this is not a winning strategy, so theremust be a play against it such that in the end there is a condition r ≤ p withπ0n(r) ≤ pn for all n ∈ ω. Let τn ∈ Mn be the names produced during thatcounterplay, and let z be the ω-sequence of ordinals obtained. The definitionsshow that for all n ∈ ω, π0n(r) Pn τn = z. It follows that z ∈Mn for all n ∈ ω,and therefore r, z are as required in (2).


The main feature of choice-coherent sequences of models we use later is thefollowing theorem connecting them with orbit equivalence relations:

Theorem 4.3.6. Let 〈Mn : n ∈ ω〉 be a generic choice-coherent sequence ofmodels. Let E be an orbit equivalence relation on a Polish space X with code inMω =

⋂nMn. If a virtual E-class is represented in Mn for every n ∈ ω, then

it is represented in Mω.

Note that a virtual E-class is an equivalence class of E-pins. Thus, the theoremsays that if there are pairwise equivalent E-pins 〈Pn, τn〉 ∈ Mn for all n ∈ ω,then there is an E-pin equivalent to them in the intersection model.

Proof. Let Γ be a Polish group continuously acting on the space X, inducingthe equivalence relation E. Let d be a compatible right-invariant metric on Γ.Let 〈P0, τ0〉 ∈ M0 be an E-pin which has an equivalent in the model Mn forevery n ∈ ω. Let λ be a cardinal so large that for each n ∈ ω, M0 is a genericextension of Mn by a poset of size < λ, and Mn contains an E-pin on a posetof size < λ equivalent to the pin 〈P0, τ0〉.

Let PΓ be Cohen forcing on the Polish group Γ, with its name γgen for thegeneric point. Let γ ∈ Γ be a Cohen-generic point, H ⊂ P0 be a generic filterand K ⊂ Coll(ω, λ) be a generic filter, mutually generic over M0; let x0 = τ0/H.In the model M0[γ,H,K], form the model N as the class of all sets hereditarilydefinable from γ · x0 and parameters in Mω. The model N is an intermediatemodel of ZFC between Mω and M0[γ,H], so by Fact 1.7.6, the model N is aforcing extension of Mω. We will argue that N and M0[H] are mutually genericextensions of Mω.

First note that this will prove the theorem. Let Q, τ ∈Mω be a poset and aname and L ⊂ Q be a filter generic over the model M0[H] such that Mω[L] = Nand τ/L = γ · x0. By the forcing theorem in the model M0, there have to beconditions p ∈ H and q ∈ L such that 〈p, q〉 τ0 E τ . It is immediate that τ asa name on Q � q is E-pinned, and the E-pin 〈Q � q, τ〉 is equivalent to 〈P0, τ0〉.This confirms the conclusion of the theorem.

To argue that N and M0[H] are mutually generic extensions of Mω, we usethe criterion of Proposition 1.7.8. In other words, if a ∈ M0[H] and b ∈ N aredisjoint subsets of some ordinal κ, we must find a set c ∈ Mω of ordinals suchthat a ⊂ c and b∩c = 0. Towards this end, move back to the model M0. Supposethat O ⊂ Γ is a nonempty open set, p ∈ P0 is a condition, a is a P0-name fora set of ordinals, and φ is a formula with parameters in Mω such that in theposet PΓ × P0, 〈O, p〉 Coll(ω, λ) ∀β ∈ a φ(β, γgen · τ0) holds. Due to thedefinition of the model N , it will be enough to find a set c ∈Mω and a condition〈O′, p′〉 ≤ 〈O, p〉 which forces a ⊂ c and Coll(ω, λ) ∀β ∈ c φ(β, γgen · τ0) holds.

Finally, we are in a position to use some coherence arguments. Let ≺ be acoherent well-ordering of M0∩Vλ; i.e. such that the restriction of ≺ to each Mn

belongs toMn. We will use the ordering to perform some coherent constructions.A typical construction of a coherent sequence (in the sense of Definition 4.2.1)proceeds by induction on n ∈ ω. If 〈vn : n ∈ ω〉 is coherent, w0 ∈ M0, and φ issome formula with parameters in Mω, one can select the ≺-least wn+1 ∈Mn+1


such that Mn |= φ(vn, wn, wn+1) if it exists; then, the sequence 〈wn : n ∈ ω〉 iscoherent. The routine details of these constructions will be suppressed below.

Find a coherent sequence 〈Pn, τn : n ∈ ω〉 of pairwise equivalent E-pins onposets in Vλ starting with 〈P0, τ0〉; i.e. for every number n ∈ ω it is the casethat Pn × Pn+1 σn E σn+1. Find a coherent sequence 〈γn : n ∈ ω〉 such thatfor each n ∈ ω, γn is a Pn×Pn+1-name for an element of the group Γ such thatτn = γn · τn+1. Let D ⊂ Γ be a fixed countable dense set in the model Mω, andlet δ0 ∈ D and ε > 0 be such that the open d-ball B(δ0, ε) ⊂ Γ is a subset of theopen set O. Find a coherent sequence 〈pn, δn : n ∈ ω〉 such that p0 ≤ p, pn ∈ Pn,δn ∈ D, and in the poset Pn × Pn+1, 〈pn, pn+1〉 d(δn · γn, δn+1) < ε · 2−n−3.Let On = B(δn, ε/2). The point of these definitions is the following claim:

Claim 4.3.7. Let n > 0. The condition 〈pi : i ≤ n〉 forces in the product∏i≤n Pi the following:

1. On ⊂ B(δ0, ε) · γ0γ1 . . . γn−1;

2. B(δ0, ε/4) · γ0γ1 . . . γn−1 ⊂ On.

Proof. Use the right invariance of the metric d to argue by induction on i ∈ nthat d(δi+1, δ0γ0γ1 . . . γi) is forced to be smaller than ε · Σj≤i2−j−3. In conclu-sion, d(δn, δ0γ0γ1 . . . γn−1) is forced to be smaller than ε/4. The two items thenfollow immediately by the right invariance of the metric d again.

Now, for every number n ∈ ω, in the model Mn form the set cn = {β ∈ κ :in the poset PΓ × Pn, 〈On, pn〉 Coll(ω, λ) φ(β, γgen · τn)}. Finally, letc = lim supn cn = {β ∈ κ : ∃∞n β ∈ cn}. It is immediate that the sequence〈cn : n ∈ ω〉 is coherent and therefore the set c belongs to the model Mω. LetO′ = B(δ0, ε/4) and p′ = p0. The following two claims stated in the model M0

complete the proof of the theorem.

Claim 4.3.8. In the poset P0, p′ a ⊂ c.

Proof. Let p′′ ≤ p′ be a condition and β ∈ κ an ordinal such that p′′ β ∈ a.It will be enough to show that for all n > 0, β ∈ cn. To this end, fix a numbern > 0 and let 〈Hi : i ≤ n〉 be a tuple of filters on the respective posets Pimutually generic over the model M0 such that pi ∈ Hi and moreover p′′ ∈ H0.Write xi = τi/Hi and γi = γi/Hi, Hi+1; so x0 = γ0γ1 . . . γn−1xn.

Let γ ∈ On be a point PΓ-generic over the model M0[Hi : i ≤ n]. Letγ′ = γγ−1

n−1γ−1n−2 . . . γ

−10 . By the invariance of the meager ideal on Γ under right

translations, γ′ ∈ Γ is a point Cohen generic over the model M0[Hi : i ≤ n]. ByClaim 4.3.7(1), γ′ ∈ B(δ0, ε) ⊂ O; moreover, γ · xn = γ′ · x0.

Let K ⊂ Coll(ω, λ) be a filter generic over the model M0[Hi : i ≤ n][γ].The model M0[Hi : i ≤ n][γ][K] is a Coll(ω, λ)-extension of both M0[γ′, H0]and Mn[γ,Hn] by the choice of λ and Fact 1.7.12. By the forcing theoremin the model M0 and the initial assumptions on the name a and the formulaφ, M0[Hi : i ≤ n][γ][K] |= φ(β, γ′ · x0). By the forcing theorem in the modelMn, the filter on PΓ × Pn given by γ,Hn must contain a condition forcing


Coll(ω, λ) φ(β, γ · τn). However, γ,Hn were arbitrary generics meeting thecondition 〈On, pn〉, so it must be the case that this condition forces Coll(ω, λ) φ(β, γgen · τn). This means that β ∈ cn as required.

Claim 4.3.9. In the poset PΓ×P0, for every ordinal β ∈ c, 〈O′, p′〉 Coll(ω, λ) φ(β, γ · τ0).

Proof. Find a number n > 0 such that β ∈ cn. Let Hi ⊂ Pi for i ≤ n befilters mutually generic over M0 containing the conditions pi respectively, withp′ ∈ H0. Write xi = τi/Hi and γi = γi/Hi, Hi+1; so x0 = γ0γ1 . . . γn−1xn.

Let γ′ ∈ O′ be a point PΓ-generic over the model M0[Hi : i ≤ n]. Letγ = γ′γ0γ1 · γn−1; by the invariance of the meager ideal on Γ under righttranslations, γ ∈ Γ is a point Cohen generic over the model M0[Hi : i ≤ n]. ByClaim 4.3.7(2), γ ∈ On; moreover, γ · xn = γ′ · x0.

Let K ⊂ Coll(ω, λ) be a filter generic over the model M0[Hi : i ≤ n][γ]. Themodel M0[Hi : i ≤ n][γ′][K] is a Coll(ω, λ)-extension of both M0[γ′, H0] andMn[γ,Hn] by Fact 1.7.12. By the forcing theorem in the model Mn and thedefinition of the set cn, M0[Hi : i ≤ n][γ′][K] |= φ(β, γ · xn). By the forcingtheorem in the model M0, the filter on PΓ × P0 given by γ′, Hn must containa condition forcing Coll(ω, λ) φ(β, γgen · τ0). However, γ′, H0 were arbitrarymeeting the condition 〈O′, p′〉, so it must be the case that this condition forcesColl(ω, λ) φ(β, γgen · τn) as required.

Example 4.3.10. Consider the equivalence relation E1 on X = (2ω)ω; it iswell-known not to be reducible to any orbit equivalence relation [48, Theorem11.8.1]. The conclusion of Theorem 4.3.6 fails for E1. To see this, choose anypartial order Q which adds a new point y ∈ 2ω. Let P be the full supportproduct of ω-many copies of Q, and let Pn be the product of the copies of thecopies of Q indexed by natural numbers ≥ n. The posets Pn for n ∈ ω naturallyform a coherent sequence. Let G ⊂ P be a generic filter, and for each n ∈ ω letGn ⊂ Pn be the restriction of G to conditions in Pn. Theorem 4.3.5 shows that〈V [Gn] : n ∈ ω〉 is a choice-coherent sequence of models, and that

⋂n V [Gn]

contains no new reals compared to V . In V [Gn], let xn ∈ X be the sequencedefined by letting xn(m) be the zero sequence if m ∈ n and the evaluation ofy by the n-th coordinate of the generic filter G otherwise. It is clear that thepoints xn all represent the same E1-class, which is not represented in V andtherefore in

⋂n V [Gn].

As a final remark, in general it is necessary to consider virtual E-classes asopposed to just E-classes in the statement of Theorem 4.3.6. To see this, startwith the trivial coherent sequence 〈Mn = V : n ∈ ω〉 and let Pn be the countablesupport product of copies of the poset Coll(ω, 2ω) indexed by natural numbers≥ n, with the natural projections from Pn to Pm added. This is a diagonallycomplete sequence of posets as in Example ??, and by Theorem 4.2.8 it inducesa choice-coherent sequence of models 〈Mn[Gn] : n ∈ ω〉 such that the model


⋂nMn[Gn] contains only ground model ω-sequences of ordinals. Now, every

model Mn[Gn] contains an enumeration of the set (2ω ∩V ) by natural numbers,and all of these enumerations are F2-related. Clearly, there is no F2-equivalentof them in the intersection model. Yet, there is a virtual F2-class related tothese enumerations in the intersection model, and even in the ground model V .


Part II

Balanced extensions of theSolovay model

117

Chapter 5

Balanced Suslin forcing

5.1 Virtual conditions

We look at the class of Suslin posets from an angle quite distinct from thestandard treatment in [7]; in particular, the center of attention is on σ-closedSuslin forcings as opposed to c.c.c. or proper forcings adding reals. Recall:

Definition 5.1.1. A preorder 〈P,≤〉 is Suslin if there is a Polish space X suchthat

1. P is an analytic subset of X;

2. the ordering ≤ is an analytic subset of X2;

3. the incompatibility relation is an analytic subset of X2.

It is important to understand that Suslin forcings may fail to be separative.Recall that conditions p0, p1 are called inseparable if for every condition q ≤ p0,q, p1 have a lower bound, and vice versa, for every condition q ≤ p1, q, p0 havea lower bound. The inseparability relation EP is an equivalence; the partialorder P is separative if this equivalence relation is the identity. A simple yetinformative example of a nonseparative Suslin forcing is in order. Consider theposet of countable partial functions from 2ω to 2 ordered by reverse extension.As stated, it is not a Suslin forcing, since it is not an analytic subset of aPolish space. One has to make an innocuous adjustment: P is in fact the setof all functions from ω to 2ω × 2 whose range is a function, and order P bysetting q ≤ p if rng(p) ⊆ rng(q). This adjustment sacrifices the separability forSuslinness. Note that the equivalence relation EP is unpinned in this case.

Throughout the rest of the book, we will make use of virtual conditionsin Suslin posets. Similar to virtual equivalence classes, these are conditionswhich may not exist in the present model of set theory and appear only in somegeneric extension, and yet we have a sensible calculus for dealing with them inthe ground model. We want the space of virtual conditions not to depend on a

119

120 CHAPTER 5. BALANCED SUSLIN FORCING

particular presentation of a given Suslin poset, and to be rich enough to harvestcertain critical features. This leads to the following definition:

Definition 5.1.2. Let 〈P,≤〉 be a Suslin forcing.

1. Let Q be a poset and τ a Q-name for an analytic subset of P . We saythat the name τ is P -pinned if Q×Q Στleft = Στright in the completionof the separative quotient of the poset P . If τ is P -pinned, then the pair〈P, τ〉 is called a P -pin.

2. Let 〈Q0, τ0〉 and 〈Q1, τ1〉 be P -pins. Define 〈Q0, τ0〉 ≤ 〈Q1, τ1〉 if Q0×Q1 Στ0 ≤ Στ1 in the separative quotient of the poset P , and 〈Q0, τ0〉 ≡〈Q1, τ1〉 if 〈Q0, τ0〉 ≤ 〈Q1, τ1〉 and 〈Q1, τ1〉 ≤ 〈Q0, τ0〉.

3. The virtual conditions of P are the equivalence classes of ≡.

For the last item of the above definition, we must verify that ≡ is in fact anequivalence relation. This is the content of the following propositions. As thefirst technical remark, note that for analytic sets A0, A1 ⊂ P , the statementΣA1 ≤ ΣA0 is equivalent to ∀p ∈ P ∀p1 ∈ A1 p ≤ p1 → ∃q ≤ p ∃p0 ∈A0 q ≤ p0. Since the ordering ≤ on P is analytic, this is a Π1

2 statement andtherefore absolute among all generic extensions by the Shoenfield absoluteness.As another remark, since any two generic extensions are mutually generic witha third, 〈Q, τ〉 is a P -pin just in case in every forcing extension V [H] and everypair of filters G0, G1 ⊂ Q in V [H] separately generic over V , Στ/G0 = Στ/H1

holds in the completion of the separative quotient of P . Similarly, it is possibleto restate the definitions of ≤ and ≡ without mutual genericity.

Proposition 5.1.3. Let 〈P,≤〉 be a Suslin forcing. Let 〈Q0, τ0〉, 〈Q1, τ1〉 beposets and names for analytic subsets of P . The following statements are abso-lute among all forcing extensions:

1. 〈Q0, τ0〉 is a P -pin;

2. 〈Q1, τ1〉 ≤ 〈Q0, τ0〉;

3. 〈Q0, τ0〉 ≡ 〈Q1, τ1〉.

Proof. We will prove (2), the other items are parallel. Suppose first that〈Q1, τ1〉 ≤ 〈Q0, τ0〉 holds, and V [H] is a generic extension of V ; we must verifythat the ≤-relation transfers to V [H]. To see this, suppose that G0 ⊂ Q0, G1 ⊂Q1 are filters mutually generic over V [H], and let A0 = τ0/G0 and A1 = τ1/G1.Thus, A0, A1 ⊂ P are analytic sets with codes in V [G0] and V [G1], respectively.By the assumption, V [G0, G1] |= ΣA1 ≤ ΣA0 in the completion of the separa-tive quotient of the poset P . This Π1

2 statement transfers to V [H][G0, G1] bythe Shoenfield absoluteness. In conclusion, V [H][G0, G1] |= ΣA1 ≤ ΣA0 andtherefore V [H] |= 〈Q1, τ1〉 ≤ 〈Q0, τ0〉.

For the other direction, if 〈Q1, τ1〉 ≤ 〈Q0, τ0〉 fails in V and V [H] is a genericextension of V , we must show that the failure of the ≤ relation transfers to V [H].

5.1. VIRTUAL CONDITIONS 121

Fix conditions q0 ∈ Q0, q1 ∈ Q1 such that in V , 〈q0, q1〉 Q0×Q1 Στ1 6≤ Στ0.This latter property of q0, q1 readily transfers to the model V [H] by the sameShoenfield absoluteness argument as the one used in the previous paragraph.

Proposition 5.1.4. ≡ is an equivalence relation on P -pins and ≤ is an orderingon ≡-equivalence classes.

Proof. We will show that ≤ is transitive, the other statements are similar. Sup-pose that 〈Q2, τ2〉 ≤ 〈Q1, τ1〉 ≤ 〈Q0, τ0〉; we need to show that 〈Q2, τ2〉 ≤〈Q0, τ0〉 holds. Let G0 ⊂ Q0, G2 ⊂ Q2 are mutually generic filters and A2 =τ2/G2 and τ0/G0; we must show that V [G0, G2] |= ΣA2 ≤ ΣA0. Let G1 ⊂ Q1

be a filter generic over V [G0, G2] and let A1 = τ1/G1. By the assumption,V [G0, G1] |= ΣA1 ≤ ΣA0 and V [G1, G2] |= ΣA2 ≤ ΣA1. By the Shoenfield ab-soluteness, V [G0, G1, G2] |= ΣA2 ≤ ΣA1 ≤ ΣA0, in particular ΣA2 ≤ ΣA0. Byanother application of the Shoenfield absoluteness, the inequality ΣA2 ≤ ΣA0

transfers from V [G0, G1, G2] to V [G0, G2].

It is now time to exhibit some virtual conditions for familiar partial orders.

Example 5.1.5. Let P be the poset of infinite subsets of ω, ordered by in-clusion. Let F be a nonprincipal filter on ω. Let τ be a Coll(ω, F )-name forthe set of all conditions p ∈ P which diagonalize the filter F , i.e. ∀a ∈ F p \ ais finite. The name τ is P -pinned, since its valuation does not depend on thechoice of the generic filter on Coll(ω, F ). It is not difficult to see that distinctnonprincipal filters generate distinct virtual conditions.

Example 5.1.6. Let P be the poset of all countable functions from 2ω to 2,ordered by reverse inclusion. Let f be any function from 2ω to 2, perhapsuncountable. Let τ be a Coll(ω, f)-name for the set of all conditions p ∈ P suchthat f ⊂ p. Again, the name τ is P -pinned. It is clear that distinct functions fgenerate distinct virtual conditions in P .

For many Suslin orders there exist virtual conditions which can be classified bynatural combinatorial objects. In particular, this holds quite often for balancedvirtual conditions, which are introduced in the following section and are thecentral topic of this book.

Proposition 5.1.7. Suppose that 〈P,≤〉 is a σ-closed Suslin poset such thatbelow any element p ∈ P there are two incompatible ones. Then the equivalence≡ has proper class many equivalence classes.

Proof. Let h : 2<ω → P be a function such that for every t ∈ ω<ω, the conditionsh(ta0) and h(ta1) are incompatible and stronger than h(t). Let g : ω → ω2 be abijection. For every ordinal α, consider the Coll(ω, α) name τα for the set of allconditions p ∈ P such that for every n ∈ ω there is (exactly one) string t(p, n) ∈2n such that p ≤ h(t), and the binary relation g′′{n ∈ ω : t(p, n + 1)(n) = 1}is isomorphic to α. It is not difficult to see that the pair 〈Coll(ω, α), τα〉 isa P -pin; in fact the evaluation of τα yields the same analytic set no matter


what the generic filter on Coll(ω, α) is. The P -pins obtained in this way are ≡-inequivalent; in fact, for distinct ordinals α 6= β, Coll(ω, α) × Coll(ω, β) Σταand Στβ are incompatible elements of the completion of the separative quotientof P .

However, certain type of virtual conditions is entirely central to the technologydeveloped in this book, and normally allows a neat classification by naturalcombinatorial objects. These are the balanced virtual conditions of the nextsection.

5.2 Balanced conditions

This section isolates the notion of a balanced condition in a given Suslin posetand a natural equivalence of balanced conditions. It turns out that every bal-anced class is represented by a virtual condition.

Definition 5.2.1. Let 〈P,≤〉 be a Suslin forcing. Let Q be a poset and τ aQ-name for an analytic subset of P . We say that 〈Q, τ〉 is a balanced pair in Pif for all posets R0, R1 and all R0 ×Q- and R1 ×Q- names σ0, σ1 for elementsof P such that R0 × Q σ0 ≤ Στ and R1 × Q σ1 ≤ Στ , it is the case that(R0 ×Q)× (R1 ×Q) σ0, σ1 are compatible conditions in the poset P .

Note that the definition of a balanced pair depends on the Suslin poset P . TheSuslin poset is not mentioned as it will be always understood from the contextand no opportunity for confusion arises. The expressions of the type σ ≤ Στ(meaning that every condition extending σ is compatible with a condition in τ)will be shortened to σ ≤ τ below as there can be no confusion. Balanced pairsand virtual conditions are quite different things; the main topic of interest in thisbook are pairs 〈Q, τ〉 which are simultaneously balanced and virtual conditions.The following proposition restates the notion of a balanced pair in terms of thegeneric extension as opposed to the forcing relation.

Proposition 5.2.2. Let 〈P,≤〉 be a Suslin forcing. Let Q be a poset and τ aQ-name for an analytic subset of P . The following are equivalent:

1. 〈Q, τ〉 is a balanced pair;

2. whenever V [H0] and V [H1] are mutually generic extensions, G0, G1 ⊂ Qare filters generic over the ground model in the respective extensions andp0 ≤ τ/G0, p1 ≤ τ/G1 are conditions in P in the respective extensionsV [H0], V [H1], then p0, p1 are compatible in P .

The class of balanced conditions is greatly simplified by introducing the followingequivalence relation on it:

Definition 5.2.3. Let 〈P,≤〉 be a Suslin forcing. If 〈Q0, τ0〉 and 〈Q1, τ1〉 arebalanced pairs, we say that 〈Q0, τ0〉 ≡b 〈Q1, τ1〉 if for all posets R0, R1 and allR0×Q0- and R1×Q1- names σ0, σ1 for elements of P such that R0×Q0 σ0 ≤ τ

5.2. BALANCED CONDITIONS 123

and R1 × Q1 σ1 ≤ τ , it is the case that (R0 × Q0) × (R1 × Q1) σ0, σ1 arecompatible conditions in the poset P .

Proposition 5.2.4. The relation ≡b is an equivalence on balanced pairs. More-over, if 〈Q, τ〉 is a balanced pair, then

1. if Q σ ≤ τ then 〈Q, σ〉 is a balanced pair ≡b-related to 〈Q, τ〉;

2. if Q is a regular subposet of R then 〈R, τ〉 is a balanced pair ≡b-related to〈Q, τ〉.

Proof. We argue for the first sentence; items (1) and (2) are immediate conse-quences of the definition. The relation ≡b is clearly symmetric and contains theidentity; we will show that it is transitive. Let 〈Q0, τ0〉 ≡b 〈Q1, τ1〉 ≡b 〈Q2, τ2〉;it must be shown that 〈Q0, τ0〉 ≡b 〈Q2, τ2〉 follows. To this end, let R0, R2 beposets and σ0, σ2 be R0 ×Q0- and R2 ×Q2-names for conditions in P strongerthan τ0, τ1 respectively. Since 〈Q0, τ0〉 ≡b 〈Q1, τ1〉 holds, there is a (R0 ×Q0)×(R1 × Q1)-name σ1 for an element of P which is a lower bound of σ0 and τ1.Since 〈Q1, τ1〉 ≡b 〈Q2, τ2〉 holds, there is a (R0 ×Q0)× (R1 ×Q1)× (R2 ×Q2)-name η which is a lower bound of σ1 and σ2. Thus, in the long product extension(R0×Q0)× (R1×Q1)× (R2×Q2), the conditions σ0, σ2 are compatible, and byMostowski absoluteness this statement transfers to the (R0 ×Q0)× (R2 ×Q2)-extension. This completes the proof of the transitivity of ≡b.

It is now time for a simple and informative example. Let P be the poset ofcountable functions from 2ω to 2, ordered by reverse inclusion. On one hand,if f : 2ω → 2 is a total function, then 〈Coll(ω, 2ω), f〉 is a balanced pair. Onthe other hand, if 〈Q, τ〉 is a balanced pair, then for every point x ∈ 2ω inthe ground model, it must be the case that Q forces f(x) ∈ dom(τ) and infact Q has to decide the value of τ(x) as well–otherwise it would be easy toviolate the balance of the pair. Let f : 2ω → 2 be the total function given by∀x ∈ 2ω Q τ(x) = f(x) and note that 〈Q, τ〉 ≡b 〈Coll(ω, 2ω), f〉. Thus, thebalanced classes for P are exactly classified by total functions from 2ω to 2.

One of the main concerns of this book is the classification of balanced classesfor various Suslin posets 〈P,≤〉. The following theorem is the basic contributionin this direction.

Theorem 5.2.5. Let 〈P,≤〉 be a Suslin forcing. Every ≡b-class contains avirtual condition. The condition is unique up to virtual condition equivalence.

Proof. Let 〈Q, τ〉 be a balanced pair; by Proposition 5.2.4 we may assume thatτ is a name for a single element of P . Consider the poset Q′ = Coll(ω,P(Q))and the name τ ′ for the analytic set {p ∈ P : for some filter G ⊂ Q meetingall open dense sets enumerated by the Q′-generic, p = τ/G}. We will first showthat the pair 〈Q′, τ ′〉 is balanced via Proposition 5.2.2.

To this end, suppose that V [H0], V [H1] are mutually generic extensions,G′0, G

′1 ⊂ Q′ are filters in the respective extensions generic over V , and r0 ≤

τ ′/G′0 and r1 ≤ τ ′/G′1 are conditions in the poset P ; we must show that r0, r1 ∈


P are compatible conditions. By the definition of the name τ ′, in V [H0] it ispossible to find a filter G0 ⊂ Q generic over V such that p0 = τ/G0 is compatiblewith r0, and similarly in the model V [G1]. Let s0, s1 ∈ P be lower bounds ofr0 and p0 in V [G0] and r1 and p1 in V [G1] respectively. Since 〈Q, τ〉 was abalanced condition in P , the conditions s0, s1 must be compatible in P , andtheir lower bound is a lower bound of r0 and r1 as well.

To show that 〈Q, τ〉 ≡b 〈Q′, τ ′〉, use both parts (1) and (2) of Proposi-tion 5.2.4 with the intermediate pair 〈Q′, τ ′〉. To show that the pair 〈Q′, τ ′〉is a virtual condition, suppose that G′0, G

′1 ⊂ Q′ are mutually generic filters.

Looking at the model V [G0, G1], it in fact turns out that τ ′/G′0 and τ ′/G′1 arein fact equal as analytic subsets of P .

To show the uniqueness of the virtual condition 〈Q′, τ ′〉 up to the virtualcondition equivalence, suppose that 〈Q′′, τ ′′〉 is another virtual condition whichis an element of the ≡b-class of the balanced class of 〈Q, τ〉; we must showthat Q′ × Q′′ Στ ′ = Στ ′′. For the ≤ direction, suppose that G′ ⊂ Q′ andG′′ ⊂ Q′′ are mutually generic filters and r ∈ P is a condition in V [G′, G′′]which is below Στ ′/G′; we must show that it is compatible with some elementof τ ′′/G′′. Let H ⊂ Q′′ be a filter generic over the model V [G′, G′′]. Since thepairs 〈Q′, τ ′〉 and 〈Q′′, τ ′′〉 come from the same balanced class, the condition rmust be compatible with every element of τ ′′/H. Let s ∈ V [G′, G′′, H] be suchthat s ≤ r and s ≤ Στ ′′/H. Since the pair 〈Q′′, τ ′′〉 is a virtual condition, thecondition s ≤ Στ ′′/H must be compatible with some condition of τ ′′/G′′. Bythe Mostowski absoluteness between the models V [G′, G′′] and V [G′, G′′, H], r iscompatible with some element of τ ′′/G′′ in the model V [G′, G′′] as required.

Question 5.2.6. Let 〈P,≤〉 be a Suslin forcing. Is it necessarily the case thatthe equivalence relation ≡b has only set many classes? Is it necessarily the casethat every balance class has a representative on a poset of size < iω1

?

It is now time to state the central definition of this book.

Definition 5.2.7. Let P be a Suslin poset. P is balanced if for every conditionp ∈ P there is a balanced virtual condition below p.

A definition of this sort immediately raises a question: which Suslin posets arebalanced? We should immediately douse the flames of entirely misguided hopes:

Proposition 5.2.8. The following Suslin posets do not have any balanced vir-tual conditions and therefore are not balanced:

1. nonatomic c.c.c. posets;

2. nonatomic tree posets;

3. posets of the form P(ω)/I where I is a countably separated Borel ideal onω.


p

��

p0 p1

q

V[H1] V[H0]

V

Figure 5.1: A balanced virtual condition p ≤ p.


Here, a tree poset on a Polish space X is an analytic family of closed subsets ofX closed under nonempty intersections with closures of basic open sets, orderedby inclusion.

Proof. For (1), suppose towards a contradiction that 〈P,≤〉 is a Suslin c.c.c.poset and 〈Q, τ〉 is a balanced pair. For every maximal antichain A ⊂ P , itsmaximality is a coanalytic statement and therefore absolute to the Q-extension.Thus, Q τ is compatible with some element of A. The balance of τ immedi-ately shows that there can be only one element of A with which τ is compatible,and the largest condition in Q identifies this element. It follows that the set{p ∈ P : Q τ ≤ p in the separative quotient of P} is a filter on P which meetsall maximal antichains, an impossibility in nonatomic posets.

For (2), suppose towards a contradiction that 〈P,≤〉 is a tree poset on aPolish space X and 〈Q, τ〉 is a balanced pair. For every basic open set O ⊂ X,the statement τ ∩ O 6= 0 must be decided by the largest condition in Q by thebalance of τ . This decision cannot be positive for two disjoint basic open subsetsof X by the balance of τ again. Thus, τ would have to be forced by Q to bea singleton (even a specific singleton in the ground model), an impossibility innonatomic tree posets.

For (3), suppose towards a contradiction that I is a countably separatedideal on ω as witnessed by a countable separating set A ⊂ P(ω), meaning thatfor every b ∈ I and c /∈ I there is a ∈ A such that b ∩ a = 0 and c ∩ a /∈ I.Suppose that 〈Q, τ〉 is a balanced pair in the poset P(ω)/I; so τ is a namefor an I-positive subset of ω. By the balance of τ , for every set a ∈ A itmust be decided by the largest condition in Q whether τ ∩ a ∈ I or not. LetB = {a ∈ A : Q τ ∩ a /∈ I}. Use the balance of the name τ to argue that anyintersection of finitely many elements of B is an infinite set. Let b ⊂ ω be aninfinite set such that for every a ∈ B, b \ a is finite. Use the density of the idealI to argue that thinning out the set b if necessary, we may assume that b ∈ I.It is now immediate that no set a ∈ A can separate the I-small set b from theI-positive set τ in the Q-extension: if a ∈ A \B then τ ∩ a ∈ I is forced, and ifa ∈ B then b ∩ a 6= 0. This is a contradiction.

Most balanced Suslin posets used in this book are σ-closed. There are σ-closedposets which are not balanced, such as the one which adds a maximal almost dis-joint family in P(ω) by countable approximations, cf. Theorem 14.1.1. Thereare some posets which are balanced and in ZFC even collapse ℵ1, cf. The-orem 8.7.2. Even such posets are valuable; remember that they prove theirworth in the choiceless symmetric Solovay extension. The balanced status ofcertain posets is nonabsolute, see for example Theorem 8.1.19 or 8.5.5. Thus,even though typically the balanced conditions correspond to traditional objectsof combinatorial set theory, the balanced status is a complicated matter. Thereis only one general preservation theorem, which is nevertheless extremely usefulfor obtaining consistency results:


Theorem 5.2.9. Let P =∏n Pn be a countable support product of Suslin

forcing notions. Balanced conditions in P are exactly classified by sequences〈pn : n ∈ ω〉 where for every n ∈ ω, pn is a balanced condition in Pn.

Proof. On one hand, if 〈Qn, τn〉 are balanced pairs for each n ∈ ω, the name forthe sequence 〈τn : n ∈ ω〉 in the product Q =

∏nQn is balanced for the poset

P essentially by the definitions. The choice of the support in the product Q isimmaterial. On the other hand, if Q is a poset and τ = 〈τn : n ∈ ω〉 is a balancedQ-name for the poset P , it must be the case that each of the names τn for n ∈ ωis balanced for the poset Pn. It equally easy to see that equivalent balancednames for P give equivalent balanced names on each coordinate, and sequencesof balanced names on the posets Pn which are coordinatewise equivalent yieldequivalent balanced names for the product forcing.

Corollary 5.2.10. The countable support product of balanced Suslin forcings isbalanced.

One issue that is constantly present in this book is the lack of absolutenessof the notions surrounding balance. As long as Question 5.2.6 remains open,it will also be necessary to relativize the definition of a balanced poset to Vκwhere κ is an inaccessible cardinal. One may think that with suitable largecardinal hypothesis on κ, one could use reflection to show that relativization isunnecessary. The best result we have in this direction is

Proposition 5.2.11. Let 〈P,≤〉 be a Suslin forcing. Let κ be a strong cardinal.The following are equivalent:

1. P is balanced;

2. Vκ |= P is balanced.

Proof. The argument uses two simple absoluteness claims of independent inter-est:

Claim 5.2.12. If M is a transitive model of ZFC, 〈Q, τ〉 is a pair in M andthe pair is balanced in V , then it is balanced in M .

Proof. Immediate by Mostowski absoluteness between generic extensions of Mand V .

Claim 5.2.13. If M is a transitive model of ZFC, 〈Q, τ〉 is a pair in M , M |=〈Q, τ〉 is balanced, and P(Q) ⊂M , then the pair 〈Q, τ〉 is balanced in V .

Proof. Work in V . Suppose that the conclusion fails, as witnessed by posetsR0, R1 and names σ0, σ1 on R0×Q and R1×Q respectively. Take an elementarysubmodel N of a large enough structure such that Q ⊂ N and |N | = |Q|. Theposets R0∩N,R1∩N and names σ0∩N and σ1∩N still witness the failure of ofthe balance of the pair 〈Q, τ〉: if G0 ⊂ (R0∩N)×Q and G1 ⊂ (R1∩N)×Q aremutually generic filters, p0 = σ0/G0 and p1 = σ1/G1 ∈ P , then N [G0, G1] |=


p0, p1 ∈ P are incompatible conditions by the forcing theorem applied in themodel N , and by the Mostowski absoluteness between N [G0, G1] and V [G0, G1],this is still true in V [G0, G1]. Now, the assumption P(Q) ⊂ M shows that theposets R0 ∩N,R1 ∩N and names σ0 ∩N and σ1 ∩N have isomorphic copies inthe model M , obtaining the failure of balance of the pair 〈Q, τ〉 in M .

Now, Claim 5.2.13 applied to M = Vκ immediately yields the implication(2)→(1). For the converse, suppose that P is balanced, p ∈ P , and find abalanced pair 〈Q, τ〉 below p. We have to deal with the unseemly possibilitythat |Q| > κ holds. Use the large cardinal hypothesis to find an elementaryembedding j : V → M with critical point κ such that 〈Q, τ〉 ∈ M ∩ Vj(κ).Claim 5.2.12 shows that M |= 〈Q, τ〉 is a balanced pair below p. By elemen-tarity of the embedding j, there must be a balanced pair 〈Q′, τ ′〉 ∈ Vκ belowp. Applying Claim 5.2.12 again with M = Vκ, we see that Vκ |= 〈Q′, τ ′〉 is abalanced pair below p. Since the condition p ∈ P was arbitrary, (2) follows.

5.3 Weakly balanced Suslin forcing

There is an interesting generalization of balanced Suslin forcing which can realizeadditional effects in extensions of the symmetric Solovay model. The basicdefinitions can be stated as a minor variation of the work done in the previoussections.

Definition 5.3.1. Let P be a Suslin forcing. A pair 〈Q, τ〉 is weakly balanced ifQ forces τ to be an analytic subset of P , and whenever R0, R1 are posets, σ0, σ1

are R0×Q and R1×Q-names for elements of P below Στ and 〈r0, q0〉 ∈ R0×Qand 〈r1, q1〉 ∈ R1 × Q are conditions then in some forcing extension there arefilters H0 ⊂ R0 × Q and H1 ⊂ R1 × Q which are separately generic over V ,〈r0, q0〉 ∈ H0, 〈r1, q1〉 ∈ H1, and σ0/H0, σ1/H1 are compatible elements of P .

Definition 5.3.2. Let P be a Suslin forcing and 〈Q0, τ0〉 and 〈Q1, τ1〉 are weaklybalanced pairs. Say that the pairs are equivalent and write 〈Q0, τ0〉 ≡wb 〈Q1, τ1〉if whenever R0, R1 are posets, σ0, σ1 are R0×Q0 and R1×Q1-names for elementsof P below τ0 and τ1 respectively and 〈r0, q0〉 ∈ R0×Q0 and 〈r1, q1〉 ∈ R1×Q1

are conditions then in some forcing extension there are filters H0 ⊂ R0×Q0 andH1 ⊂ R1×Q1 which are separately generic over V , 〈r0, q0〉 ∈ H0, 〈r1, q1〉 ∈ H1,and σ0/H0, σ1/H1 are compatible elements of P .

Proposition 5.3.3. The relation ≡wb is an equivalence relation on weakly bal-anced pairs.

Proof. The relation is clearly symmetric and contains the identity as a sub-set. For the transitivity, let 〈Q0, τ0〉, 〈Q1, τ1〉 and 〈Q2, τ2〉 be weakly balancedpairs such that 〈Q0, τ0〉 ≡wb 〈Q1, τ1〉 and 〈Q1, τ1〉 ≡wb 〈Q2, τ2〉. To show that〈Q0, τ0〉 ≡wb 〈Q2, τ2〉 holds, let R0, R2 be posets and σ0, σ2 be R0 × Q0- andR2 × Q2-names for elements of P which are forced to be smaller than someelement of τ0 and τ2 respectively, and let 〈r0, q0〉 and 〈r2, q2〉 be conditions

5.3. WEAKLY BALANCED SUSLIN FORCING 129

in the two product posets. Let κ be a cardinal bigger than |P(R0 × Q0)|.Since 〈Q0, τ0〉 ≡wb 〈Q1, τ1〉 holds, an absoluteness argument shows that inthe Coll(ω, κ) × Q0-extension, there is a filter H0 ⊂ R0 × Q0 generic overthe ground model containing 〈r0, q0〉 such that σ0/H0 and τ1 as conditionsin P have a lower bound; call the Coll(ω, κ) × Q0-name for the bound σ1.Since 〈Q1, τ1〉 ≡wb 〈Q2, τ2〉 holds, in some forcing extension there are filtersH1 ⊂ Coll(ω, κ) × Q1 and H2 ⊂ R2 × Q2 separately generic over the groundmodel, such that 〈r2, q2〉 ∈ H2 and σ1/H1, σ2/H2 are compatible elements ofP . Let H0 = H0/H1 ⊂ R0 ×Q0; this is a filter generic over the ground model,containing the condition 〈r0, q0〉 such that σ0/H0 ≤ σ1/H1.

In total, the conditions σ0/H0, σ2/H2 ∈ P are compatible, and the relation〈Q0, τ0〉 ≡wb 〈Q2, τ2〉 has been verified.

The notion of a weakly balanced pair is a faithful extension of the notion of abalanced pair. This is the content of the following proposition.

Proposition 5.3.4. Let P be a Suslin forcing.

1. Every balanced pair is weakly balanced;

2. the class of balanced pairs is invariant under the ≡wb equivalence;

3. the relations ≡b and ≡wb coincide on balanced pairs.

Proof. (1) is immediate from the definitions. For (2), suppose towards a contra-diction that 〈Q0, τ0〉 and 〈Q1, τ1〉 are ≡wb-equivalent pairs and 〈Q0, τ0〉 is bal-anced, while 〈Q1, τ1〉 is not. The latter statement is witnessed by some posetsR0, R1, names σ0, σ1, and conditions 〈r0, q0〉 ∈ R0 ×Q1 and 〈r1, q1〉 ∈ R1 ×Q1

which force in the product (R0 ×Q0)× (R1 ×Q1) that σ0, σ1 are incompatibleelements of P .

Use the ≡wb-equivalence assumption to find posets S0, S1 and S0, S1-namesG0, H0, K0 and G1, H1, K1 respectively so that S0 G0 ⊂ Q0 and H0 × K0 ⊂R0 × Q1 are filters generic over V such that r0 ∈ H0, q0 ∈ K0, and τ0/G0,σ0/H0×K0 are conditions compatible in P , with a lower bound χ0 ∈ P . Similarobjects exist on the S1-side.

Now, let L0 ⊂ S0, L1 ⊂ S1 be filters mutually generic over V . The balanceof the pair 〈Q0, τ0〉 shows that χ0/L0, χ1/L1 are compatible conditions in theposet P . Consider the filters H0 = H0/L0 ⊂ R0, K0 = K0/L0 ⊂ Q1, andH1 = H1/L1 ⊂ R0, K1 = K1/L1 ⊂ Q1. The filters H0 × K0 ⊂ R0 × Q1 inV [L0] and H1 × K1 ⊂ R1 × Q1 in V [L1] are generic over V . Since the filtersL0, L1 are mutually generic, so are H0 ×K0 and H1 ×K1 by Corollary 1.7.9.At the same time, the conditions σ0/H0 ×K0 and σ1/H1 × K1 in P must becompatible, because they are weaker than the compatible conditions χ0/L0 andχ1/L1 respectively. This is a contradiction with the initial choice of σ0, σ1.

(3) is proved in a similar way; the argument is left to the patient reader.

Proposition 5.3.5. Let P be a Suslin forcing. Every ≡wb-class contains avirtual condition, which is unique up to ≡-equivalence.


Proof. Let 〈Q, τ〉 be a weakly balanced condition; strengthening τ if necessary,we may assume that τ is in fact a name for an element of P . Let κ be an ordinalsuch that P(Q) ⊂ Vκ and let σ be a Coll(ω, Vκ)-name for the set {p ∈ P : ∃G ⊂Q G is a filter meeting all the dense subsets of Q in V Vκ such that p = τ/G}.It is clear that the pair 〈Coll(ω, Vκ), σ〉 is a P -pin. We will show that the pair〈Coll(ω, Vκ), σ〉 is ≡wb-related to 〈Q, τ〉.

Suppose that R0, R1 are posets and σ0, σ1 are R0×Coll(ω, Vκ)- and R1×Q-names for elements of P stronger than σ and τ respectively, and 〈r0, q0〉, 〈r1, q1〉are conditions in the respective products. Without loss of generality we mayassume that there is a cardinal λ > |Vκ| such that R0 = Coll(ω, λ). Let K be theR0 × Coll(ω, Vκ)-name for a filter on Q which is generic over V and such thatσ0 ≤ τ/K is forced. By abstract forcing theory, strengthening the condition〈r0, q0〉 if necessary, we may cast the product R0 ×Coll(ω, Vκ) below 〈r0, q0〉 asthe poset Coll(ω, λ) × Q below some condition 〈r′0, q′0〉 so that the name K isa name for the generic filter on the second coordinate of the product. By theweak balance of the name τ , in some generic extension there are filtersH ′0 ⊂Coll(ω, λ)×Q and H1 ⊂ R1 ×Q containing the conditions 〈r′0, q′0〉 and 〈r1, q1〉such that the condition σ/H ′0, σ1/H1 are compatible in P . The filter H ′0 canbe cast as a filter H0 ⊂ R0 × Coll(ω, Vκ) meeting the condition 〈r0, q0〉. Thiscompletes the proof of the ≡wb-equivalence.

For the uniqueness part, suppose that 〈Q0, τ0〉 and 〈Q1, τ1〉 are P -pins whichare both ≡wb-related to 〈Q, τ〉; we must show that they are ≡-related. Supposetowards a contradiction that this fails. Then, in the Q1 × Q0-extension, theremust be an element p ∈ P which is below some element of τ0 and incompatiblewith any element of τ1 (or vice versa). By a Mostowski absoluteness argument,this element p will maintain its incompatibility property in every further forcingextension. Let σ0 be a Q1 × Q0-name for this element, let σ1 be a Q1-namefor any element of τ1, and observe that 〈Q1 ×Q0, σ0〉 and 〈Q1, σ1〉 witness thatthe the pairs 〈Q0, τ0〉 and 〈Q1, τ1〉 are ≡wb-unrelated, contradicting the initialassumptions.

Unlike the balanced conditions, the weakly balanced virtual conditions can beactually recognized in the ordering of virtual conditions by a natural first orderproperty.

Proposition 5.3.6. Let P be a Suslin forcing. Let 〈Q, τ〉 be a P -pin. Thefollowing are equivalent:

1. 〈Q, τ〉 is weakly balanced;

2. 〈Q, τ〉 is an atom in the ordering of virtual conditions.

Recall that an element p of a partial order is an atom if every element compatiblewith p is in fact above p.

Proof. For (1)→(2) direction, let 〈R, σ〉 be a P -pin which is not above 〈Q, τ〉; wemust show that it is incompatible with 〈Q, τ〉. Suppose towards a contradictionthat it is compatible, with a lower bound 〈S, χ〉. In the R × Q extension, the

5.3. WEAKLY BALANCED SUSLIN FORCING 131

inequality Στ ≤ Σσ must fail, so there is a condition p0 for an element of Pwhich is below Στ but incompatible with Σσ. In the S×R×Q-extension, thereis a condition p1 of P which is below Σχ, and therefore also below Σσ and Στ .Use the weak balance of the pair 〈Q, τ〉 to find, in some generic extension, filtersH0×G0 ⊂ R×Q and K1×H1×G1 ⊂ S×R×Q separately generic over V suchthat the conditions p0/H0×G0 and p1/K1×H1×G1 are compatible in P , witha lower bound p. Note that the sums Σσ/H0 and Σσ/H1 in the completion ofthe poset P must coincide, as 〈R, σ〉 is a P -pin. However, the condition p shouldbe incompatible with the former and below the latter by the forcing theorem.This is a contradiction.

For the (2)→(1) direction, suppose that 〈Q, τ〉 is a P -pin which is an atomin the ordering of virtual conditions. To prove the weak balance, suppose thatR0, R1 are posets, σ0, σ1 are R0 × Q and R1 × Q-names for elements of Pstronger than τ , and let 〈r0, q0〉 and 〈r1, q1〉 be conditions in the products. Tofind the instrumental generic filters, let κ be an ordinal such that P(R0×Q) andP(R1 × Q) are both subsets of Vκ, and consider the Coll(ω, Vκ)-names χ0, χ1

for analytic subsets of P defined by χ0 = {p ∈ P : ∃G ⊂ R0 ×Q such that G isa filter meeting all open dense subsets of R0 ×Q in V Vκ such that 〈r0, q0〉 ∈ Gand p = σ/G}. The name χ1 is defined in the same way.

It is not difficult to see that 〈Coll(ω, Vκ), χ0〉 and 〈Coll(ω, Vκ), χ1〉 are bothP -pins. They are also both ≤ 〈Q, τ〉 by their definition. By the assumption onthe P -pin 〈Q, τ〉, they must both be ≡-equivalent to 〈Q, τ〉. This means that inthe Coll(ω, Vκ)-extension, there must be conditions p0 ∈ χ0 and p1 ∈ χ1 whichare compatible in P . Reviewing the definition of the names χ0 and χ1, we getthe filters G0 ⊂ H0×K0 and G1 ⊂ H1×K1 separately generic over V such that〈r0, q0〉 ∈ G0, 〈r1, q1〉 ∈ G1, and σ0/G0 and σ1/G1 are compatible conditions inP as desired.

Finally, we record the central definition of this section.

Definition 5.3.7. A Suslin poset P is weakly balanced if below every conditionp ∈ P there is a weakly balanced virtual condition stronger than p.


Chapter 6

Simplicial complex forcings

6.1 Basic concepts

Many examples of σ-closed Suslin partially ordered sets in this book are pre-sented in the same way:

Definition 6.1.1. Let X be a set.

1. A set K of finite subsets of X is a simplicial complex if it is closed undersubset;

2. a set A ⊂ X is a K-set if [A]<ℵ0 ⊂ K. It is maximal if it is not a propersubset of another K-set;

3. the poset PK ⊂ Xω consists of countable K-sets ordered by reverse inclu-sion;

4. Agen is the PK-name for the union of all sets in the generic filter.

Unless specifically stated otherwise, we will tacitly assume that every singletonbelongs to K. When the simplicial context K is understood from the context, weput P = PK in this section. The poset P is obviously σ-closed. By an elementarydensity argument, the set Agen is forced to be a maximal K-set. The poset Pcan be naturally presented as a Suslin forcing by replacing the countable K-setswith their enumerations by natural numbers, and replacing the reverse inclusionordering by reverse inclusion of ranges of the enumerations. We will neglect thisinnocuous step in this section as it merely complicates the notation.

Nearly every poset considered in this book can be presented as a poset ofthe form PK for a Borel simplicial complex K on a Polish space X. Namely,for a poset Q let K be the simplicial complex of the finite subsets of Q whichhave a common lower bound. Under suitable assumptions on definability andexistence of lower bounds (which are invariably satisfied), the posets Q andPK are naturally forcing equivalent. However, this point of view rarely bringsany new insight. In this chapter, we deal with simplicial complexes that are

133

134 CHAPTER 6. SIMPLICIAL COMPLEX FORCINGS

in some way algebraically natural, and their algebraic structure leads to theclassification of the balanced conditions. We discovered a number of possibilities.Many of them are categorized by the properties of the following central conceptsassociated with any simplicial complex whatsoever.

Definition 6.1.2. Let K be a Borel simplicial complex on a Polish space X.ΓK is the Borel graph on K consisting of all pairs 〈a, b〉 where a, b ∈ K, a∪b /∈ K,and a, b are inclusion minimal such: if a′ ⊆ a and b′ ⊆ b then a′ ∪ b′ /∈ K iffa′ = a and b′ = b.

In particular, if 〈a, b〉 ∈ ΓK then any other disjoint partition of the set a∪ b intononempty pieces c ∪ d must have c, d ∈ K and 〈c, d〉 ∈ ΓK.

Definition 6.1.3. Let K be a Borel simplicial complex on a Polish space X.

1. A K-walk is a finite sequence 〈ai : i ∈ j〉 of elements of K such that forevery k ∈ j \ {0} there is a set b ⊂

⋃i∈k ai such that 〈ak, b〉 ∈ ΓK;

2. A K-walk 〈ai : i ∈ j〉 is said to start at a point x0 ∈ X if a0 = {x0} andreach a point x1 ∈ X if for some i ∈ j x1 ∈ ai;

3. EK is the set of all pairs 〈x0, x1〉 such that there is a K-walk starting fromx0 and reaching x1 and a K-walk starting from x1 and reaching x0.

Proposition 6.1.4. Let K be a Borel simplicial complex on a Polish space X.EK is an analytic equivalence relation. It is the smallest equivalence relation Eon X such that for every finite set a ⊂ X, a ∈ K if and only if for every E-classc, a ∩ c ∈ K.

Proof. Write Γ = ΓK and E = EK. It is immediate that the relation E is anequivalence. The complexity assertion is also immediate from the definitions.For the optimality assertion, for an equivalence relation F on X write φ(F )for the statement that for every finite set a ⊂ X, a ∈ K if and only if forevery F -class c, a ∩ c ∈ K. We first need to verify that φ(E) holds. Supposetowards a contradiction that it does not, and let a ⊂ X be an inclusion-minimalcounterexample. Note that for every E-class c such that a∩c 6= 0 it must be thecase that 〈a∩c, a\c〉 ∈ Γ. Now let x ∈ a be any point and let c be the E-class ofx. Produce a K-walk from x which reaches all the finitely many points in c∩ a.The walk can be extended by the set a \ c, reaching an arbitrary element of a.This means that all points of a belong to the same E-class, a contradiction.

Now suppose that F be an equivalence relation on X for which φ(F ) holds;we must argue that E ⊆ F . Suppose towards a contradiction that this failsand let x0, x1 ∈ X be E-related points which are not F -related. Let 〈ai : i ∈ j〉be a K-walk starting from x0 and reaching x1. Let k ∈ j be the smallestnumber such that ak 6⊂ [x0]F . There must be a set b ⊂

⋃i∈k ai such that

〈b, ak〉 ∈ Γ. Applying φ(F ) to the union ak ∪ b, we see that it must be the casethat (ak ∪ b) ∩ [x0]F /∈ K. Thus, considering the proper subset a′k = ak ∩ [x0]F ,we get b ∪ a′k /∈ K, contradicting the assumption that 〈b, ak〉 ∈ Γ.

6.2. LOCALLY COUNTABLE COMPLEXES 135

To conclude this section, we mention two operations on simplicial complexes.Both of them may seem trivial from the simplicial complex view, but theygenerate quite unpredictable operations on the associated posets PK. Severalclasses of complexes introduced below are nearly trivially closed under theseoperations.

Definition 6.1.5. Let K be a Borel simplicial complex on a Polish space X,and let B ⊂ X be a Borel set. The restriction K � B is just the set K∩ [B]<ℵ0 .

Definition 6.1.6. Let X be a Polish space and for each n ∈ ω, let Kn be aBorel complex on a Borel set Bn ⊂ X. The product K =

∧nKn is the simplicial

complex of all finite sets a ⊂ X such that ∀n a ∩Bn ∈ Kn.

One way to use the latter operation appears is when the Borel sets Bn forn ∈ ω are pairwise disjoint. In this case, the poset PK corresponds to thecountable support product of the posets PKn for n ∈ ω. However, if the Borelsets Bn overlap, the forcing effects of this operation become much more difficultto analyze.

6.2 Locally countable complexes

In this section, we discuss the simples class of simplicial complexes to handle.

Definition 6.2.1. Let K be a Borel simplicial complex on a Polish space X.We say that K is locally countable if the graph ΓK is locally countable.

It follows from a routine application of the Lusin–Novikov theorem that if aBorel simplicial complex K on a Polish space X is locally countable, then EK is acountable Borel equivalence relation. In particular, a Borel simplicial complex Kis locally countable if and only if there is a countable Borel equivalence relation Eon X such that for every finite set a ⊂ X, a ∈ K if and only if a∩c ∈ K for everyE-class c. The classification of balanced pairs in the case of locally countablesimplicial complexes is particularly simple: they correspond to maximal K-sets.This is the content of the following theorem:

Theorem 6.2.2. Let K be a locally countable Borel simplicial complex on aPolish space X.

1. For every maximal K-set A ⊂ X, the pair 〈Coll(ω,X), A〉 is balanced inthe poset P = PK;

2. for every balanced pair 〈Q, τ〉, there is a maximal K-set A ⊂ X such thatthe balanced pairs 〈Q, τ〉 and 〈Coll(ω,X), A〉 are equivalent;

3. distinct maximal K-sets yield inequivalent balanced pairs.

In particular, the poset P is balanced.


Proof. For (1), suppose that V [H0], V [H1] are mutually generic extensions of Vand p0 ∈ V [H0], p1 ∈ V [H1] are conditions extending A; we must show thatp0 ∪ p1 is a K-set. Suppose towards a contradiction that a0 ⊂ p0 and a1 ⊂ p1

are finite sets such that a0 ∪ a1 /∈ K. Choosing a0, a1 inclusion-minimal, weconclude that {a0, a1} ∈ ΓK. Since the graph ΓK is locally countable, by theMostowski absoluteness between V [H0] and V [H0, H1] it must be the case thatas a0 ∈ V [H0], a1 ∈ V [H0] must hold as well. Thus, a1 ∈ V [H0] ∩ V [H1] = V .Since A was a maximal K-set and A∪ a1 ⊂ p1 is a K-set, it follows that a1 ⊂ Aand a0 ∪ a1 ⊂ p0, contradicting the assumption that a0 ∪ a1 /∈ K.

For (2), let 〈Q, τ〉 be a balanced pair; strengthening τ if necessary we mayassume that it is a Q-name for a single K-set. Replacing Q with Q×Coll(ω,X)and strengthening the condition τ if necessary, we may assume that Q ∀x ∈XV x ∈ τ or there is a ⊂ τ such that {x} ∪ a /∈ K. For each point x ∈ X,either Q x ∈ τ or Q ∃a ⊂ τ ∈ ΓK and a ⊂ τ by a balance argument. LetA ⊂ X be the set of all points x ∈ X for which the first option prevails. ThenQ A = τ ∩V , so A is a K-set; in view of (1) and Proposition 5.2.4, it is enoughto show that A is a maximal K-set. Let x ∈ X\A. Then Q ∃a ⊂ τ {x}∪a /∈ K;choosing a inclusion minimal, we conclude that Q {{x}, a} ∈ ΓK. By the localcountability of the graph ΓK and a Mostowski absoluteness between the groundmodel and its Q-extension, it must be that Q a ⊂ V and then a ⊂ A. Itfollows that A ∪ {x} is not a K-set as desired.

Finally, (3) is obvious. For the last sentence, any condition p ∈ P canbe extended to a maximal K-set, which then represents a balanced conditionstronger than p by (1).

Example 6.2.3. Let E be a countable Borel equivalence relation on a Polishspace X. Let K be the set of all finite sets consisting of pairwise E-unrelatedelements. Then ΓK consists of all edges {{x0}, {x1}} where x0, x1 ∈ X aredistinct E-related points, so K is locally countable. The balanced pairs areclassified by maximal K-sets, which are precisely the E-transversals.

For the following example, recall the notion of a perfect matching of a graph;this is just a set of edges such that each vertex gets exactly one edge in the setadjacent to it. For a locally finite bipartite graph Γ, the existence of a perfectmatching is equivalent to Hall’s marriage condition [38]: for every finite set a ofvertices on one side of the bipartition, the set of neighbors of a has cardinalityat least that of a.

Example 6.2.4. Let Γ be a locally finite bipartite Borel graph on a Polishspace X satisfying the Hall’s marriage condition; view Γ as a subset of [X]2.Let K be the simplicial complex of all finite subsets of Γ which can be completedto a perfect matching. Then K is a Borel locally countable simplicial complex:given a finite set a ⊂ Γ, the membership of a in K can be detected by checkingall fragments of a in the components of Γ which are countable. The balancedpairs are classified by maximal K-sets which are precisely the perfect matchingsof Γ.

6.3. COMPLEXES OF BOREL COLORING NUMBER ℵ1 137

For the following example, recall the notion of an end of an infinite connectedgraph [36]. If the graph in question is in addition acyclic, an end can be identifiedwith an orientation of the graph in which every vertex gets exactly one edgeflowing out of it.

Example 6.2.5. Let Γ be a locally finite acyclic Borel graph on a Polish spaceX with all components infinite; view Γ as a symmetric subset of X2. Let K bethe simplicial complex of all finite subsets a ⊂ Γ consisting of finite sets a suchthat for each Γ-path-connectedness class c ⊂ X the set a∩c2 can be extended toan end. It is immediate that K is a Borel locally countable simplicial complex.The balanced conditions are classified by maximal K-sets, which are orientationsof the graph Γ in which every vertex has exactly one point in its outflow.

Example 6.2.6. Let Γ be a locally countable Borel graph on a Polish space Xof chromatic number ≤ n for some finite n ∈ ω. Let K be the simplicial complexof all finite partial Γ-colorings by at most n colors which can be completed toa total coloring by at most n colors. Then K is a locally countable simplicialcomplex: given a finite coloring a, the membership of a in K can be detected bychecking all fragments of a in the components of Γ which are countable. Thebalanced pairs are classified by maximal K-sets which are precisely the totalΓ-colorings.

6.3 Complexes of Borel coloring number ℵ1

The next class of examples is obtained as a generalization of the locally countablesimplicial complex using a definable version of a concept due to Erdos and Hajnal[25].

Definition 6.3.1. 1. Let Γ be a Borel graph on a Polish space X. We saythat the graph has Borel coloring number ℵ1 if there is a Borel directedgraph ~Γ (an orientation of Γ) such that for each edge {x0, x1} ∈ Γ, either

〈x0, x1〉 ∈ ~Γ or 〈x1, x0〉 ∈ ~Γ, and the ~Γ-outflow of each point is countable .

2. Let K be a Borel simplicial complex on a Polish space X. We say that Khas Borel coloring number ℵ1 if the graph ΓK has Borel coloring numberℵ1.

In particular, every locally countable Borel simplicial complex has Borel coloringnumber ℵ1. A Mostowski absoluteness argument shows that if a Borel graph hasBorel coloring number ℵ1 then this feature will persist to all generic extensions.

To prove the balance for the posets associated with Borel simplicial com-plexes of Borel coloring number ℵ1 and to classify the balanced conditions, weneed a technical weakening of maximality.

Definition 6.3.2. Let K be a Borel simplicial complex on a Polish space X.A K-set A ⊂ X is weakly maximal if for every point x ∈ X either x ∈ A or forevery countable set b ⊂ X there is c ∈ K disjoint from b such that (A∩ b)∪ c isa K-set, but (A ∩ b) ∪ c ∪ {x} is not.


In particular, every maximal K-set is weakly maximal. For a weakly maximalset A ⊂ X, let τA be the Coll(ω,X)-name for the analytic set of all p ∈ P suchthat for all x ∈ X ∩ V , if x ∈ A then x ∈ p and if x /∈ A then p ∪ {x} is not aK-set. In particular, if A is a maximal K-set, then Coll(ω,X) ΣτA = A.

Theorem 6.3.3. Let K be a Borel simplicial complex on a Polish space X ofBorel coloring number ℵ1.

1. For every weakly maximal K-set A ⊂ X, the pair 〈Coll(ω,X), τA〉 is bal-anced in the poset P = PK;

2. for every balanced pair 〈Q, τ〉, there is a weakly maximal K-set A ⊂ Xsuch that the balanced pairs 〈Q, τ〉 and 〈Coll(ω,X), τA〉 are equivalent;

3. distinct weakly maximal K-sets yield inequivalent balanced pairs.


Proof. For (1), we must first show that Coll(ω,X) τA 6= 0. By a Mostowskiabsoluteness argument, it is just enough to produce some generic extension inwhich there is a K-set p ⊂ X such that for all x ∈ X ∩ V , if x ∈ A thenx ∈ p and if x /∈ A then p ∪ {x} is not a K-set. To do this, let Q be theposet of nonstationary subsets of [X]ℵ0 and let j : V → M be the Q-namefor the associated generic ultrapower. In particular, M is an ω-model of ZFCcontaining j′′X as an element, represented by the identity function; this setis forced to be countable in M . Now, consider a product of copies Qx of Qindexed by elements x ∈ X \ A, with mutually generic filters Gx ⊂ Qx andin the model V [Gx : x ∈ X \ A] consider the generic ultrapower embeddingsjx : V → Mx. By the weak maximality of A and elementarity, there is a setcx ∈ Mx disjoint from V such that A ∪ cx is a K-set, and A ∪ cx ∪ {x} isnot. We claim that p = A ∪

⋃x cx is a K-set; this will complete the proof.

Suppose towards a contradiction that this is not the case, and find finite setsa ⊂ A and ax ⊂ cx for x in some finite index set I such that a ∪

⋃x ax /∈ K

and ax are inclusion-minimal possible. Pick x0 be any element of I such thatax06= 0, let b = ax0

, and let c = a ∪⋃x6=x0

ax. It is clear that {c, d} ∈ Γ.

Let ~Γ be a Borel orientation of Γ with countable outflows, and assume fordefiniteness that 〈c, b〉 ∈ ~Γ. By a Mostowski absoluteness argument betweenV [Gx : x 6= x0] and V [Gx : x ∈ X \ A], it must be the case that b, as much as

all other points of the countable ~Γ-outflow of c, belongs to V [Gx : x 6= x0], sob ∈ V [Gx : x 6= x0] ∩ V [Gx0 ] = V . This contradicts the initial choice of the setsax ⊂ cx.

Now, for the balance part, suppose that V [H0] and V [H1] are mutuallygeneric extensions of V , and p0 ∈ V [H0] and p1 ∈ V [H1] are conditions suchthat for all x ∈ X ∩ V , if x ∈ A then x ∈ p0 and if x /∈ A then p0 ∪ {x} is nota K-set, and similarly for p1. We must show that p0 ∪ p1 is a K-set. Supposetowards a contradiction that this fails, and let a0 ⊂ p0 and a1 ⊂ p1 be finite setssuch that a0 ∪ a1 /∈ K. Passing to an inclusion-minimal example of this form

6.4. MODULAR COMPLEXES 139

if necessary, we may assume that {a0, a1} ∈ Γ. Let ~Γ be a Borel orientation of

Γ with countable outflows, and assume for definiteness that 〈a0, a1〉 ∈ ~Γ. Bya Mostowski absoluteness argument between V [H0] and V [H0, H1], it must be

the case that a1, as much as all other points of the countable ~Γ-outflow of a0,belongs to V [H0], so a1 ∈ V [H1]∩V [H0] = V . It follows that a1 ⊂ A ⊂ p0, andthis contradicts the assumption that a0 ∪ a1 /∈ K.

For (2), let 〈Q, τ〉 be a balanced pair; strengthening τ if necessary we mayassume that it is a Q-name for a single K-set. Replacing Q with Q×Coll(ω,X)and strengthening the condition τ if necessary, we may assume that Q ∀x ∈XV x ∈ τ or there is a ⊂ τ such that {x} ∪ a /∈ K. For each point x ∈ X,either Q x ∈ τ or Q ∃a ⊂ τ ∈ ΓK and a ⊂ τ by a balance argument. LetA ⊂ X be the set of all points x ∈ X for which the first option prevails. ThenQ A = τ ∩ V and therefore A is a K-set; in view of (1) and Proposition 5.2.4,it is enough to show that A is a weakly maximal K-set. Let x ∈ X \ A andb ⊂ X be countable. Observe that Q ∃c ⊂ τ {x} ∪ c /∈ K; note that for suchset c, (A ∩ b) ∪ c is still a K-set. Use the Mostowski absoluteness argument tofind a set c with these properties in the ground model; this confirms that A isa weakly maximal K-set.

Finally, (3) is obvious. For the last sentence, any condition p ∈ P canbe extended to a maximal K-set, which then represents a balanced conditionstronger than p by (1).

Example 6.3.4. Let g : 2ω → (2ω)ℵ0 be any Borel function. Let K be thesimplicial complex of g-free sets on X = 2ω. It is immediate that K has Borelcoloring number ℵ1: the graph ΓK consists of all pairs {x0, x1} such that eitherx0 ∈ f(x1) or x1 ∈ f(x0). The Borel orientation of ΓK with countable outflowscan be defined by considering which of the two cases occurs. To consider a casein which the classification of balanced conditions is particularly simple, supposethat every point in (2ω)ℵ0 has uncountable preimage. Then, any g-independentset A ⊂ X is weakly maximal: for every countable set b ⊂ X and every pointx ∈ X \ A there is an uncountable set of points y ∈ X such that x ∈ g(y)and g(y) ∩ (A ∩ b) = 0, and uncountably many such points y can be found alsooutside of the countable set

⋃{g(z) : z ∈ A ∩ b}, making the set (A ∩ b) ∪ {y}

free, while (A ∩ b) ∪ {y} ∪ {x} is not.

6.4 Modular complexes

Another class of simplicial complexes in this section is not as closely tied to thegraph ΓK. The definition is motivated by concerns of geometric model theory.

Definition 6.4.1. A Borel simplicial complex K on a Polish space X is modularif there is a Borel modular function f : K → [Y ]ℵ0 for some Polish space Y suchthat

1. (monotonicity) f(0) = 0 and a ⊂ b implies f(a) ⊂ f(b);


2. (modularity) for all a, b ∈ K, a∪b ∈ K if and only if f(a)∩f(b) = f(a∩b).

For a K set A ⊂ X we set f(A) =⋃{f(a) : a ∈ [A]<ℵ0}.

Note that the properties of the function f are coanalytic, and therefore transferto any transitive model of set theory by the Mostowski absoluteness, in par-ticular to all generic extensions. It is not easy to see which Borel simplicialcomplexes are modular and which are not. As the simplest initial example, letX = R viewed as a vector space over the rationals, and let K be the simplicialcomplex on X consisting of finite linearly independent sets. The simplicial com-plex is modular as witnessed by the modular function f which to each finite setof reals assigns its linear span.

The balanced virtual conditions in the poset P = PK for modular complexesK are again classified by certain K-sets, as recorded in the following definition:

Definition 6.4.2. Let K be a simplicial complex on a Polish space X. A K-setA ⊂ X is strongly maximal if for every Borel collection B ⊂ K at least one ofthe following occurs:

1. there are a0 6= a1 in B such that a0 ∪ a1 ∈ K;

2. there is b ∈ B such that b ⊂ A;

3. there is a countable set p ⊂ A such that for no b ∈ B, p ∪ b is a K-set.

Fittingly, a strongly maximal set A ⊂ X must be a maximal K-set. For this,given x ∈ X consider the set B = {{x}} and consult the three options ofDefinition 6.4.3. The first option is impossible, the second option yields x ∈ A,and the third option shows gives that A ∪ {x} is not a K-set. The maximalityof the set A follows.

Theorem 6.4.3. Suppose that K is a modular Borel simplicial complex on aPolish space X and write P = PK.

1. Whenever A ⊂ X is a strongly maximal set then the pair 〈Coll(ω,X), A〉is balanced in P ;

2. every balanced pair is equivalent to 〈Coll(ω,X), A〉 for some strongly max-imal set A ⊂ X;

3. distinct strongly maximal sets give rise to inequivalent balanced pairs.

In particular, under the Continuum Hypothesis the poset P is balanced.

We do not know if the poset P is balanced in ZFC, even though this does occurin all specific cases under consideration in this book.

Proof. Let f : K → [Y ]ℵ0 be a Borel modular function. For (1), suppose thatV [H0] and V [H1] are mutually generic extensions of the ground model, anda0 ∈ V [H0] and a1 ∈ V [H1] are elements of K such that A ∪ a0, A ∪ a1 are


K-sets; we must show that a0 ∪ a1 ∈ K. Suppose towards a contradiction thatthis fails. Using the maximality of the set A, enlarge the sets a0, a1 so thata0∩a1 = a0∩V = a1∩V . The modularity of the function f implies that the setf(a0) ∩ f(a1) \ f(a0 ∩ a1) is nonempty, containing some element y ∈ Y . Sincey ∈ V [H0] ∩ V [H1], the product forcing theorem implies that y ∈ V .

Back in V , consider the set B ⊂ K, B = {b ∈ K : y ∈ f(b) and for noproper subset c ⊂ b, y ∈ f(c)}. Apply Definition 6.4.2 to A,B. The firstoption is impossible by the modularity of the function f . The second optiongives a set b ∈ K such that b ⊂ A and y ∈ f(b). Increasing the set b ifnecessary, we may arrange that a0 ∩ V ⊂ b. But then, y ∈ f(a0) ∩ f(b) whiley /∈ f(a0∩ b) = f(a0∩V ) = f(a0∩a1). The modularity of the function f showsthat b∪a0 /∈ K and so A∪a0 is not a K-set, contradicting the initial asumptions.In the third option of the trichotomy, there is (in V ) a countable set p ⊂ A suchthat no element of B can be added to it. Let b ⊂ a0 be an inclusion-minimalsubset of a0 such that y ∈ f(b). Then b ∈ B and p ∪ b is a K-set, contradictingthe choice of p. (1) has been proved.

For (2), assume that 〈Q, τ〉 is a balanced pair. Replacing Q with Q ×Coll(ω,X) and strengthening τ if necessary, we may assume that τ is in fact aname for an element of PK, and Q ∀x ∈ X ∩ V x ∈ τ or {x} ∪ τ is not aK-set. By the balance, for each x ∈ X it has to be the case that Q x ∈ τ orQ {x} ∪ τ is not a K-set; let A ⊂ X be the set of all points x ∈ X for whichthe former alternative occurs. We will show that A ⊂ X is a strongly maximalK-set. Then, 〈Q, τ〉 is equivalent to 〈Coll(ω,X), A〉. Since Q τ ≤ A holds,in view of (1) and Proposition 5.2.4 〈Q, τ〉 is equivalent to 〈Coll(ω,X), A〉 asdesired.

Now, it is clear that A is a K-set, since it is forced to be a subset of theK-set τ . Suppose towards a contradiction that A ⊂ X is not strongly maximal,as witnessed by some Borel set B ⊂ K.

Claim 6.4.4. There is a poset R and an R-name b such that R b ∈ K andA ∪ b is a K-set.

Proof. Let R be the poset of all stationary subsets of [X]ℵ0 ordered by inclusion.Now, if G ⊂ R is a generic filter and j : V →M is the associated generic ultra-power, the set X ∩V belongs to M and it is countable in M , as it is representedby the identity function. Thus, A = j(A) ∩ (X ∩M) is a countable subset ofj(A) in the model M , and by the failure of the third option in Definition 6.4.2in V and elementarity of j, M |= ∃b b ∈ B and A ∪ b is a K-set. Since M is anω-model of set theory, these two statements about b transfer without change tothe generic extension V [G].

The treatment now divides into two cases.Case 1. For some condition q ≤ Q and a Q-name b for an element of B, q τ∪bis a K set. In this case, note that since the second option in Definition 6.4.2 failsfor A,B, it must be the case that b is forced not to be a subset of the groundmodel. Let H0, H1 ⊂ Q be mutually generic filters meeting the condition q. Thesets b/H0, b/H1 ∈ B must be distinct by the product forcing theorem. Since


the first option of Definition 6.4.2 fails, it must be that b/H0∪ b/H1 /∈ K. Thus,τ/H0 ∪ b/H0 and τ/H1 ∪ b/H1 are incompatible conditions in the poset P inthe respective models V [H0] and V [H1] contradicting the balance of the pair〈Q, τ〉.Case 2. Q ∀b ∈ B τ ∪ b is not a K-set. Let R and b be a poset and aname as in the claim. Note that Q×R τ ∪ b is not a K-set by the Mostowskiabsolutenes between the Q- and Q × R-extension. By passing to a conditionin Q and R if necessary, find a Q-name c and a finite set d ⊂ A such thatQ c ⊂ τ, d = c ∩ V , and Q×R c ∪ b /∈ K. Note that R d ∪ b ∈ K. By themodularity of the function f , Q×R f(d∪ b)∩ f(c \ V ) 6= 0 holds, and by theproduct forcing theorem, all elements in the intersection must be in the groundmodel. In particular, there must be a point y ∈ Y and a condition q ∈ Q forcingy ∈ f(c\V ). Let H0, H1 ⊂ Q be filters mutually generic over the ground model,containing the condition q ∈ Q. Then y ∈ f(c/H0 \ V ) ∩ f(c/H1 \ V ) whiley /∈ f(c/H0 ∩ H1 \ V ) = f(0) = 0. The modularity of the function f shows thatc/H0 ∪ c/H0 \ V /∈ K, in particular τ/H0, τ/H1 are incompatible conditions inthe poset P . This contradicts the balance of the pair 〈Q, τ〉.

(2) has just been proved. (3) is obvious. For the last sentence, assume thatthe Continuum Hypothesis holds and let p ∈ P be an arbitrary condition. Let〈Bα : α ∈ ω1 enumerate all Borel subsets of K. By recursion on α ∈ ω1 constructa descending chain of conditions pα ∈ P so that p0 = p, pα =

⋃β∈α pβ for limit

ordinals α, and pα+1 contains some element ofBα as a subset if such an extensionof pα exists at all. Then

⋃α∈ω1

pα is a strongly maximal K-set which by (1)yields a balanced virtual condition stronger than p.

The following examples provide several interesting classes of modular complexes.

Example 6.4.5. Every locally countable Borel simplicial complex is modular.

Proof. Let K be a locally countable Borel simplicial complex on a Polish spaceX. Consider the function f : K → (ΓK)ℵ0 defined by {c, d} ∈ f(a) just in caseeither c ⊆ a or d ⊆ a. The values of the function f are countable sets by thelocally countable assumption on the simplicial complex; it is not difficult to seethat the function f is Borel in a suitable sense. The monotonicity of the functionf is immediate; this leaves us with verifying the modularity of the function f .

Suppose that a, b ∈ K are finite sets. Suppose first that a ∪ b ∈ K and{c, d} ∈ f(a)∩f(b); we need to show that {c, d} ∈ f(a)∩f(b). By the definitionof the function f , either c ⊆ a or d ⊆ a, and c ⊆ b or d ⊆ b must hold. Now,c ⊆ a and d ⊆ b is impossible since c ∪ d /∈ K while a ∪ b ∈ K. For the samereason, the conjunction d ⊆ a and c ⊆ b is impossible as well. This means thatone of c, d must be a subset of both a and b, so of a ∩ b. The definition of thefunction f then shows that {c, d} ∈ f(a ∩ b) as required.

Now, suppose that a∪b /∈ K; we must find an edge {c, d} ∈ f(a)∩f(b) whichdoes not belong to f(a∩ b). To this end, find an inclusion-minimal set e ⊂ a∪ bsuch that e /∈ K, and let c = e∩ a and d = e \ a. The minimal choice of e showsthat {c, d} ∈ ΓK. Since c ⊂ a and d ⊂ b, it is clear that {c, d} ∈ f(a) ∩ f(b).


Since e 6⊂ a and e 6⊂ b, it follows that neither c nor d is a subset of a ∩ b, so{c, d} /∈ f(a ∩ b) as required.

Many simplicial complexes that we consider, and in particular modular simpli-cial complexes, are in fact matroids. For completeness, we include the definitionof matroids and several basic facts.

Definition 6.4.6. [2, Chapters VI and VII] A simplicial complex K on a set Xis a matroid if for any sets a, b ∈ K, if |b| > |a| then there is a point x ∈ b \ asuch that a ∪ {x} ∈ K.

It is immediate that for a Borel simplicial complex K on a Polish space X,the statement that K is a matroid is coanalytic and therefore absolute amongtransitive models of set theory. Matroid theory introduces several useful basicnotions that apply to every matroid. For a set a ⊂ X let rk(a) be the largestcardinality of a set b ⊂ a such that b ∈ K, if such exists; otherwise rk(a) =∞.For a set a ⊂ X let a = {x ∈ X : rk(a) = rk(a ∪ {x})}. This is a closureoperation satisfying the properties from the following fact, which may serve asan alternative definition of a matroid [2, 6.9].

Fact 6.4.7. Let K be a matroid on a set X and let a, b ⊂ X be finite sets. Then

1. a ⊂ a;

2. (idempotence) b ⊆ a implies b ⊆ a;

3. (exchange principle) if x ∈ a ∪ {y} \ a then y ∈ a ∪ {x}.

The set X with the closure operation satisfying the above properties is a pre-geometry. For a finite set a ⊂ X, a ∈ K just in case for any set b ⊂ a and anypoint x ∈ a \ b, x /∈ b holds–K is the complex of free sets of the pre-geometry.The closed sets (the sets a ⊂ X such that a = a) form a lattice under inclusion;the pre-geometry is modular if the lattice of its closed sets is modular. Thiscan be equivalently characterized by x ∈ a ∪ b iff there are points xa ∈ a andxb ∈ b such that x ∈ {xa, xb}. The pre-geometry is locally countable if theclosure of any finite set is countable. The following example justifies our choiceof terminology.

Example 6.4.8. Let K be a Borel matroid on a Polish space X whose pre-geometry is locally countable and modular. Then K is a modular simplicialcomplex and the closure operation is a modular function on K. Every maximalK-set is strongly maximal.

Proof. Let f : K → [X]ℵ0 be the closure operation; it is immediate from itsdefinition that it is a Borel function. To verify the modularity of f , supposethat a, b ∈ K are finite sets. Suppose first that f(a) ∩ f(b) 6= f(a ∩ b) and workto show that a ∪ b is free. Let x ∈ X be a point in f(a) ∩ f(b) which is not inf(a ∩ b). Let a′ ⊂ a be inclusion minimal superset of a ∩ b such that x ∈ f(a′).Note that a′ 6⊂ a ∩ b, choose a point y ∈ a′ \ (a ∩ b), and let a′′ = a′ \ {y}. By


the exchange property of the pre-geometry, it follows that y ∈ f(a′′ ∪ {x}). Bythe monotonicity and idempotence of the pre-geometry, y ∈ f(a′′ ∪ b) and sincey ∈ a∪ b and y /∈ a′′ ∪ b, this means that a∪ b /∈ K. This direction does not usethe modularity assumption on the pre-geometry.

Now suppose that a ∪ b /∈ K and work to show that f(a) ∩ f(b) 6= f(a ∩ b).Let c be an inclusion minimal subset of a∪b such that a∩b ⊂ c, c is free, and forsome x ∈ a∪ b, c∪{x} is not free; pick the witness x. First argue that x ∈ f(c).To this end, let d ⊂ c∪{x} be minimal such that there is y ∈ (c∪{x})\d whichis in f(d). If x /∈ d then y must be equal to x by the freeness of the set c; andif x ∈ d, then by the exchange property x ∈ f(d∪ {y} \ {x}), so x ∈ f(c) again.

Now, assume for definiteness that x ∈ b. Let a′ = c \ b and b′ = c ∩ b. Sincex ∈ f(a′ ∪ b′), the modularity assumption on the pre-geometry yields pointsx0 ∈ f(a′) and x1 ∈ f(b′) such that x ∈ f(x0, x1). Note that {x0, x1} ∈ K: iffor example x1 ∈ f({x0}), then by idempotence x1 ∈ f(a′), by the exchangeproperty there would have to be some element z ∈ a′ such that z ∈ f({x1} ∪(a′ \ {z})), and by idempotence again z ∈ f(c \ {z}), contradicting the freenessof the set c. By the freeness of the set b, x /∈ f({x1}). By the exchange propertyapplied again, x0 ∈ f({x1, x}) ⊂ f(b). It follows that x0 ∈ f(a) ∩ f(b). At thesame time, x0 /∈ f(a∩b), since then (as a∩b ⊂ c holds) x0 ∈ f(b′) and thereforex ∈ f(b′), violating the freeness assumption on b.

To show that every maximal K-set is strongly maximal, let A ⊂ X be a max-imal K-set and let B ⊂ K be a set. Let M be a countable elementary submodelof a large structure containing B and A. If the third option of Definition 6.4.2fails for B, there must be a set b ∈ B such that (M ∩A)∪ b is a K-set. If b ∈Mthen b ⊂ A by the maximality of A and the elementarity of the model M andthe second option of Definition 6.4.2 holds. Suppose then that b /∈ M . By theelementarity of the model M , there must be a set a ∈ B∩M such that b∩M ⊂ a.We claim that a∪b ∈ K, confirming the first option of Definition 6.4.2. Supposetowards a contradiction that this fails; by the exchange property, there have tobe sets a′ ⊂ a, M ∩ b ⊂ b′ ⊂ b and a point x ∈ b \ b′ such that x ∈ f(a′ ∪ b′).By the modularity, there have to be points y, z such that y ∈ f(a′), z ∈ f(b′)and x ∈ f(y, z). By the exchange property, y ∈ f(z, x) ⊂ f(b). At the sametime, y ∈M and so by the maximality of the set A and the elementarity of themodel M , there has to be d ⊂ A in the model M such that y ∈ f(d). In total,y ∈ f(b) ∩ f(d), by the initial assumption on the set b b ∪ d ∈ K holds, and bythe modularity of the function f , y ∈ f(b∩d) ⊂ f(b′). Then x ∈ f(y, z) ⊂ f(b′),contradicting the freeness of the set b.

Example 6.4.9. (A linear matroid) Let X be a Borel vector space over acountable field, and let K be the matroid of finite linearly independent subsetsof X. The pre-geometry of K is well-known to be modular, so the simplicialcomplex K is modular. Let P be the poset of countable linearly independentsets. The balanced conditions in P are classified by bases of X over the field.

One important point of the present section is that there are interesting matroidsmodular in the sense of Definition 6.4.1 even though their pre-geometries arenot modular.


Example 6.4.10. (A graphic matroid) Let G be a Borel graph on a Polish spaceX. Let K be the simplicial complex on G consisting of finite acyclic subsets ofG. Then K is modular. Every maximal K-set is strongly maximal.

Proof. Consider the (non-Polishable) group Y of finite subsets of X with thesymmetric difference operation, viewed as a vector space over the binary field.The simplicial complex L of linearly independent finite subsets of Y is modularby Example 6.4.8. It is easy to see that K is just the restriction of L to G, whereevery edge of G is viewed as a two-element subset of X. The modular functionfor L restricted to L witnesses the modularity of K.

To show that every maximal K-set is strongly maximal, let A ⊂ X be a max-imal K-set and let B ⊂ K be a set. Let M be a countable elementary submodelof a large structure containing B and A. If the third option of Definition 6.4.2fails for B, there must be a set b ∈ B such that (M ∩ A) ∪ b is a K-set. Ifb ∈ M then b ⊂ A by the maximality of A and the elementarity of the modelM and the second option of Definition 6.4.2 holds. Suppose then that b /∈ M .By the elementarity of the model M , there must be a set a ∈ B ∩M such thatb ∩M ⊂ a. We claim that a ∪ b ∈ K, confirming the first option of Defini-tion 6.4.2. Suppose towards a contradiction that this fails, and let c ⊂ a ∪ b bea cycle. The cycle has to contain some edges in b \M ; let c′ ⊂ c be a maximalcontiguous part of c containing only edges in b \M . The beginning and endingvertex of c′ (denoted by v0, v1 respectively) must belong to M . Thus, v0, v1 areconnected by a G-path, they are also connected by A-path by the maximalityof A, and such a path d ⊂ A must be found in the model M by the elementarityof M . Then d ∪ c′ is a cycle in (M ∩A) ∪ b, contradicting the choice of the setb.

Example 6.4.11. (A transversal matroid) Let X be a Polish space partitionedinto Borel sets B0, B1, and let G be a Borel bipartite graph between B0 andB1. Let K be the matroid on G consisting of finite sets of edges in which notwo distinct edges share a vertex. Then K is modular. If every vertex in X hasuncountable degree, then a K-set A ⊂ G is maximal if and only if it is a perfectmatching.

Proof. To exhibit the modular function, let f(a) = {x ∈ X : there is an edgee ∈ a such that x is one of the two vertices of e}. Now suppose that every vertexin X has uncountable degree and proceed towards the classification of stronglymaximal sets. Let A ⊂ X be any K-set.

First, assume that A is strongly maximal. To see that A has to be a perfectmatching, for every vertex x ∈ X consider the set Bx = {{e} : x is one of thevertices in e}. Consider the alternatives of Definition 6.4.2 for Bx. (1) fails and(3) is impossible as the vertex x has uncountable degree in G; therefore, (2) hasto hold, meaning that the set A contains an edge with the vertex x on it and soA is a perfect matching.

Second, suppose that A is a perfect matching and B ⊂ K is a Borel set. Toverify Definition 6.4.2 for A,B, let M be a countable elementary submodel of alarge structure containing G,A,B. If the third option of Definition 6.4.2 fails,


there must be b ∈ B such that (M ∩ A) ∪ b is a matching. Let b′ = b ∩M anduse the elementarity of the model M to find a ∈ B ∩M such that b′ ⊂ a. Thereare several possibilities now. If a = b = b′ then a ∪ A is a matching by theelementarity of M , and by the maximality of A, a ⊂ A and the option (2) hasbeen verified. If a 6= b and a∪b is a matching, then option (1) follows. Otherwise,there has to be an edge e0 ∈ b connecting some vertex x mentioned in a with avertex outside of M . Since A is a perfect matching and M is elementary, thereis an edge e1 ∈ A∩M containing x; since the other vertex of e1 is in M , it mustbe the case that e0 6= e1 and so (M ∩ A) ∪ b is not a matching, contradictingthe choice of the set b.

Not every modular simplicial complex needs to be a matroid.

Example 6.4.12. Let X be a Polish space and let Γ be a Borel collectionof finite subsets of X. Let K be the simplicial complex on Γ consisting offinite subsets of Γ which consist of pairwise disjoint sets. It is immediate thatthe simplicial complex K is modular, with the modular function f(a) =

⋃a.

The simplicial complex K is typically not a matroid: for example, if there arepoints x0, x1 which are together contained in a single element of Γ and are alsocontained in respective disjoint elements of Γ, then the definitory property of amatroid immediately fails.

Finally, we present several examples of simplicial complexes which are not mod-ular. To rule out the existence of a function with the modular property, we usethe following simple criterion:

(*) there are pairwise disjoint sets {a, bα : α ∈ ω1} in K such that for anyα, β ∈ ω1 bα ∪ bβ ∈ K holds and for every α ∈ ω1, a ∪ bα /∈ K holds.

It is clear that if (*) holds, then no function f can have the modularity property:the values of f(bα) for α ∈ ω1 would have to be pairwise disjoint in view of thefirst demand in (*), and then there would have to be an ordinal α ∈ ω1 suchthat f(a) ∩ f(bα) = 0, which violates the modularity property in view of thesecond demand in (*).

Example 6.4.13. (An algebraic matroid) Let X be an uncountable Polish field,and let K be the simplicial complex on X consisting of sets algebraically freeover some countable subfield F ⊂ X. The simplicial complex K satisfies (*) andtherefore is not modular. To see this, let B ⊂ X be an uncountable algebraicallyfree set, let a ⊂ B be an arbitrary set of size 2, a = {s, t}, and for each x ∈ B \alet bx = {x+s, xt}. We will show that {a, bx : x ∈ X \a} exemplify (*). Clearly,for each x ∈ B \ a, a ∪ bx is not free. Also, for distinct elements x, y ∈ B \ a,the set bx ∪ by is free. To see that, it is enough to show that the field Q(bx, by)contains s, t, x, y and therefore has transcendence degree 4. To recover s, t, x, yfrom bx ∪ by, write w = x − y and z = xy−1; both clearly belong to Q(bx, by).Then the recovery follows from the string of equalities y = w(z − 1)−1, x = zy,t = (xt)x−1 and s = (x+ s)− x. (This elegant argument was pointed out to usby Peter Sin.)


Example 6.4.14. (A gammoid) Let X be a Polish space. Let K be the matroidon finite subsets of X containing those finite sets a ⊂ [X]<ℵ0 which allow theexistence of an injective function g : a→ X such that ∀u ∈ a f(u) ∈ u. To verifythat (*) is satisfied, select pairwise distinct points y0, y1 and xα for α ∈ ω1 ofthe underlying space X, let a = {{y0}, {y1}} and bα = {{y0, xα}, {y1, xα}}.

Example 6.4.15. Let X be an uncountable Polish space and let K be thesimplicial complex on [X]2 consisting of those finite sets a ⊂ [X]2 such that thereis no set b ∈ [X]3 such that [b]2 ⊂ a. The associated poset PK adds a maximaltriangle-free graph on X. The simplicial complex K satisfies (*); therefore, itis not modular. To see this, let x0, x1 ∈ X be any distinct points and leta = {{x0, x1}}. For each z ∈ X distinct from x0, x1 let bz = {{x0, z}, {x1, z}}.It is easy to check that {a, bz : z ∈ X} exemplify (*).

Example 6.4.16. Let X = P(ω) and let K be the simplicial complex of all finitesets a ⊂ X such that

⋂a is infinite. The associated poset PK adds a Ramsey

ultrafilter. The simplicial complex K satisfies (*) and so is not modular. To seethis, let F be any filter on ω which is not generated by countably many sets.Let a ⊂ ω be an infinite set such that the complement of a is in F , and let bαfor α ∈ ω1 be pairwise distinct elements of F disjoint from a. It is not difficultto verify that the sets {a}, {bα} : α ∈ ω1 exemplify (*).

The examples above are selected so that the associated poset PK is alwaysbalanced. Since the criterion (*) rules out an arbitrary modular function asopposed to a Borel modular function, and in general the modular functionsseem to be very difficult to construct by transfinite recursion procedures, thefollowing question suggests itself:

Question 6.4.17. Is there a Borel simplicial complex which possesses a modularfunction, but no Borel modular function?

We conclude this section with the best general result on the existence of stronglymaximal sets we can prove in ZFC.

Theorem 6.4.18. Let K be a Borel modular simplicial complex on a Polishspace X such that its modular function takes only finite values. Then everycountable K-set can be extended to a strongly maximal K-set.

Proof. Let C ⊂ X be a countable K-set. Let f : K → [Y ]<ℵ0 be a Borel modularfunction. Let κ be the cardinality of the continuum, and let {Bα : α ∈ κ} be anenumeration of all Borel subsets of K. By recursion on α ∈ κ construct K-setsCα so that

• C = C0 ⊂ C1 ⊂ . . . , |Cα| = |α|+ ℵ0;

• for every ordinal α, either there is a set b ∈ Bα such that b ⊂ Cα+1 orthere is a countable set c ⊂ Cα such that for no b ∈ Bα, c ∪ b is a K-set.


To perform the recursion, for a limit ordinal α let Cα =⋃β∈α Cβ . Now suppose

that the set Cα has been constructed and work to find Cα+1. Let M be acountable elementary submodel of a large structure containing Cα. If there isno b ∈ Bα such that (Cα ∩M) ∪ b is a K-set, then let Cα+1 = Cα and proceedto the next ordinal. If, on the other hand, there is a set b ∈ Bα such that(Cα ∩M)∪ b is a K-set, we will work to produce a set a ∈ Bα such that Cα ∪ ais a K-set. Setting Cα+1 = Cα ∪ a will complete the recursion step in this caseas well.

Use Proposition 1.7.10 to find a perfect collection {Hz : z ∈ 2ω} of filters onColl(ω,< κ) pairwise mutually generic over the model M . The product forcingtheorem guarantees that for distinct points z0, z1 ∈ 2ω, M [Hz0 ]∩M [Hz1 ] = M .Since the set Cα has size smaller than continuum, a counting argument showsthat there is a point z ∈ 2ω such that Cα ∩M [Hz] and f(Cα)∩M [Hz] are bothsubsets of M . Note that b ∩M , f(b) ∩M , and X ∩M are all sets in M [Hz]and use a Mostowski absoluteness argument for the model M [Hz] to find a seta ∈ Bα in the model M [Hz] such that a∩M = b∩M and f(a)∩M = f(b)∩M .We claim that Cα ∪ a is a K-set as desired.To see this, suppose that d ⊂ Cα is a finite set. Adding some points of Cα ∩Mto d if necessary, we may assume that f(d)∩f(a) ⊂ f(d∩M)∩f(a). The choiceof the point z then implies that f(d)∩f(a) = f(d)∩f(a)∩M = f(d∩M)∩f(b).The latter set is equal to f(d∩M ∩ b) since (d∩M)∪ b ∈ K by the initial choiceof b and the fact that f is a modular function. Finally, d∩M ∩ b = d∩ a by thechoice of the point z again, and so f(d) ∩ f(a) = f(d ∩ a) and a ∪ d ∈ K by themodularity of the function f .

Once the recursion has been performed, let A =⋃α Cα and use the second

item of the recursion demands to conclude that A ⊂ X is a strongly maximalK-set extending C.

6.5 Gδ matroids

The final class of simplicial complexes investigated in this section deals withcomplexes associated with pre-geometries of a form particularly simple fromthe descriptive set theoretic view.

Definition 6.5.1. A simplicial complex K on a Polish space X is Gδ if it is a Gδsubset of [X]<ℵ0 , where the latter set is equipped with the topology inheritedfrom the hyperspace K(X) of compact subsets of X with the Vietoris topology.

Theorem 6.5.2. Let K be a Gδ matroid on a Polish space X. Write P = PK.

1. If A ⊂ X is a maximal K-set, then the pair 〈Coll(ω,A), A〉 is balanced inthe poset P ;

2. whenever 〈Q, τ〉 is a balanced pair in P then there is a maximal K-set suchthat the balanced pairs 〈Q, τ〉 and 〈Coll(ω,A), A〉 are equivalent;

3. distinct maximal K-sets yield inequivalent balanced pairs.

6.5. Gδ MATROIDS 149


Proof. For (1), suppose that R0 σ0 ⊂ X is a finite set such that A∪σ0 is a K-set, and similarly for R1, σ1; we must show that R0×R1 σ0∪σ1 ∈ K. Supposetowards a contradiction that this fails; rearranging the names σ0, σ1 and passingto a condition in R0, R1 if necessary, we may assume that there is a finite seta ⊂ A and R0 σ0∩V = 0 and R1 σ1∩V = 0 and R0×R1 a∪σ0∪σ1 /∈ K.

Let M be a countable elementary submodel of a large enough structure,containing all objects named so far. Let R′0 = M∩R0, σ′0 = M∩σ0, R′1 = M∩R1

and σ′1 = M ∩ σ1. Perhaps more precisely, σ′0 is the name on R′0 such that forevery basic open set O ⊂ K(X) and every condition r ∈ R′0, if r R0

σ0 ∈ Othen r R′0 σ′0 ∈ O; similarly for σ′1. It is not difficult to see that σ′0 is anR′0-name for a finite subset of X disjoint from the ground model. The followingclaim is key.

Claim 6.5.3. R′0 A ∪ σ′0 is a K-set.

Proof. Suppose towards a contradiction that this fails. Then there must be afinite set b ⊂ A and a condition r ∈ R′0 such that r R′0 b ∪ σ′0 /∈ K. Bythe complexity assumption on K, strengthening the condition r if necessary wecan find a closed set C ⊂ K(X) in the model M such that C ∩ K = 0 andr b∪σ′0 ∈ C. Now, step out of the model M . The poset R0 forces b∪σ0 /∈ C;thus, there is a basic open set O ⊂ K(X) in the Vietoris topology disjoint fromC and a condition s ≤ r such that s R0 b ∪ σ0 ∈ O. The set O is of the formO = {K ∈ K(X) : K ⊂ U and K ∩ Vi 6= 0 for all i ∈ n} for some choice of basicopen sets U, Vi ⊂ X and some number n ∈ ω. Let a = {i ∈ n : b∩ Vi = 0}; thens σ0 ⊂ U and σ0 ∩ Vi 6= 0 for all i ∈ a. By the elementarity of the modelM , there is a condition t ≤ r in M such that t R0 σ0 ⊂ U and σ0 ∩ Vi 6= 0for all i ∈ a; then t R′0 σ

′0 ⊂ U and σ0 ∩ Vi 6= 0 for all i ∈ a, in other words

t R′0 b ∪ σ′0 ∈ O. This contradicts the assumption that r R′0 b ∪ σ

′0 ∈ C.

Now, let H1 ⊂ R′1 be a filter generic over the model M , in V . Let H0 ⊂ R′0be a filter generic over V . By the product forcing theorem, the filters H0 ⊂ R′0and H1 ⊂ R′1 are mutually generic over the model M . Write b0 = σ′0/H0 andb1 = σ′1/H1; so b1 ∈ V . By the elementarity of the model M and the forcingtheorem applied in it to the poset R0 × R1, M [H0, H1] |= a ∪ b1 ∈ K anda ∪ b1 ∪ b0 /∈ K. By the Mostowski absoluteness V [H0] |= a ∪ b1 ∈ K anda ∪ b1 ∪ b0 /∈ K.

By the maximality of the initial set A ⊂ X, find a finite set c ⊂ A such thata∪ b1 is a subset of the algebraic closure of c. By the claim, c∪ b0 ∈ K. Find aninclusion maximal set d ⊂ a∪ b1∪ b0 containing a∪ b1 as a subset and such thatd ∈ K. Let x ∈ (a ∪ b1 ∪ b0) \ d be an arbitrary point; clearly x ∈ b0. Considerthe pre-geometry associated with the matroid K. Denote its closure operationby algebraic closure to eliminate possible confusion with topological closure inthe space X. By the exchange property of the pre-geometry, x belongs to thealgebraic closure of d. By the idempotence of the pre-geometry, x belongs to


the algebraic closure of c ∪ (b0 ∩ d). However, this contradicts the fact thatc ∪ b0 ∈ K holds. (1) has just been proved.

For (2), let 〈Q, τ〉 be a balanced pair. Replacing Q with Q×Coll(ω,X) andstrengthening τ if necessary, we may assume that Q ∀x ∈ X ∩ V x ∈ τ or{x} ∪ τ is not a K-set. By a balance argument, for each x ∈ X it must be thecase that either Q x ∈ τ or Q {x} ∪ τ is not a K-set. Let A ⊂ X be theset of those points x ∈ X for which the former case prevails. Since Q τ ≤ A,in view of (1) and Proposition 5.2.4 it will be enough to show that A ⊂ X is amaximal K-set.

Certainly, A ⊂ X is a K-set since it is forced to be a subset of a K-set τ .Now suppose that A ⊂ X is not maximal and pick a point x /∈ A such thatA ∪ {x} is still a K-set. Passing to a condition of Q if necessary, we may finda finite set a ⊂ A and a Q-name σ for a finite subset of τ such that Q forces xto belong to the algebraic closure of a ∪ σ and not to belong to the closure ofany of its proper subsets. Let η be a Q-name for an inclusion-maximal subsetof a∪{x}∪ σ which is in K and contains a∪{x}. It is not equal to a∪{x}∪ σ,and it contains every element of σ \ η in its algebraic closure by the exchangeproperty of the pre-geometry. Let H0, H1 ⊂ Q be filters mutually generic overV and write p0 = τ/H0, b0 = σ/H0, and c0 = η/H0, and similarly for p1, b1, c1.It will be enough to show that a ∪ b0 ∪ b1 /∈ K, since this will contradict theinitial assumption of balance of the name τ and the resulting compatibility ofconditions p0, p1 ∈ P .

Note that x belongs to the algebraic closure of a ∪ b0. By the idempotenceof the pre-geometry, it follows that every element of the nonempty set b1 \ c1belongs to the algebraic closure of a ∪ b0 ∪ c1. Since a ∪ b0 ∪ c1 ⊂ a ∪ b0 ∪ b1,this shows that the latter set is not free and completes the proof of (2).

Finally, (3) is obvious. The last sentence of the theorem follows from astraighforward application of Kuratowski–Zorn lemma–every K-set can be ex-tended to a maximal K-set.

Example 6.5.4. [42] (An algebraic matroid) Let X be a Polish field over acountable subfield F . Consider the Borel simplicial complex K of finite subsetsof X which are algebraically free over F . K is well-known to be generated bythe Borel pre-geometry of algebraic closure. Also, a finite set a ⊂ X is not inK just in case it is a solution to a nontrivial polynomial with coefficients in F .There are only countably many such polynomials and for each of them, the setof solutions is Fσ. Thus, the simplicial complex K is Gδ and the poset PK isbalanced by Theorem 6.5.2. The poset adds a transcendence basis to the fieldX as the union of the generic filter.

6.6 Quotient variations

Many simplicial complex forcings are naturally connected with a Borel equiva-lence relation.

6.6. QUOTIENT VARIATIONS 151

Definition 6.6.1. Let E be a Borel equivalence relation on a Polish space X.A Borel simplicial complex K on X is an E-quotient complex if the membershipof any finite set a ⊂ X in K depends only on [a]E . The letter E is left out if Eis understood from the context or will be specified later.

In this section we identify two general schemas for defining useful quotient sim-plicial complexes and their associated posets: collapse posets for quotient car-dinals and uniformization posets for quotient spaces.

Definition 6.6.2. Let E,F be Borel equivalence relations on Polish spaces X,Ywith uncountably many classes each. Let KE,F = K be the simplicial complex onthe set (X×Y )∪Y consisting of finite sets a such that for all 〈x0, y0〉, 〈x1, y1〉 ∈ a,x1 E x0 if and only if y1 F y0 holds, and for all 〈x0, y0〉, y2 ∈ a, y0 F y2 fails.Clearly, K is an (E × F ) ∪ F -quotient complex. The associated poset PK isreferred to as the E,F -collapse poset.

If A ⊂ (X×Y )∪Y is a generic set, the part A∩ (X×Y ) is an injection fromX/E to Y/F , while the part A∩Y is the complement of the range of A∩(X×Y ).The balanced conditions are neatly classified by injections from the virtual E-quotient space X∗∗ to the virtual F -quotient space Y ∗∗. If g : X∗∗ → Y ∗∗ is aninjection, let τg be the Coll(ω,iω1)-name for the set of those elements p ∈ PKsuch that for each pair 〈c, d〉 ∈ g, there is a pair 〈x, y〉 ∈ p such that x ∈ X isa realization of the virtual E-class c and y ∈ Y is a realization of the virtualF -class d. Moreover, if d is a virtual F -class not in the range of g, then p isrequired to contain an element y ∈ Y which is a realization of the virtual F -classd.

Theorem 6.6.3. Let E,F be Borel equivalence relations on respective Polishspaces X,Y . Let P be the E,F -collapse poset.

1. For every total injection g : X∗∗ → Y ∗∗, the pair 〈Coll(ω,iω1), τg〉 is a

balanced P -pin;

2. for every balanced pair 〈Q, τ〉 there is a total injection g : X∗∗ → Y ∗∗ suchthe pair 〈Coll(ω,iω1

), τg〉 is equivalent to 〈Q, τ〉;

3. distinct injections as in (1) yield inequivalent balanced conditions.

Proof. Write R = Coll(ω,iω1). For (1), the pair 〈Coll(ω,iω1), τg〉 is clearly aP -pin. For the balance, let V [H0], V [H1] be mutually generic extensions of theground model and p0 ∈ V [H0] and p1 ∈ V [H1] be conditions in the analytic setτg/H0 = τg/H1; we must show that p0, p1 ∈ P are compatible.

To see this, suppose for example that 〈x0, y0〉 ∈ p0, 〈x1, y1〉 ∈ p1, and x0 Ex1; we must show that y0 E y1. By a mutual genericity argument, the pointsx0, x1 must be realizations of the same virtual E-class c in the ground model.Since both p0, p1 are K-sets, the definition of the name τg shows that both y0, y1

must be realizations of the virtual F -class g(c) and therefore y0 F y1 as desired.Similarly, if 〈x0, y0〉 ∈ p0 and y2 ∈ p1, we need to show that y0 F y2 fails.

Suppose towards a contradiction that y0 F y2 holds. By a mutual genericity


argument, the points y0, y2 are representations of the same virtual F -class din the ground model. Now, if d ∈ rng(g) then p1 cannot be a K-set, and ifd /∈ rng(g) then p0 cannot be a K-set. There are no other options, and thiscontradiction completes the proof of (1).

For (2), let g : X∗∗ → Y ∗∗ be the collection of all pairs 〈c, d〉 ∈ X∗∗ × Y ∗∗so that R × Q 〈x, y〉 ∈ τ for some realizations x, y of the virtual classes c, d.It will be enough to show that g is a total injection from X∗∗ to Y ∗∗ and thepair 〈Q, τ〉 is equivalent to 〈R, τg〉.

To see that dom(g) = X∗∗ and g is an injection, let c ∈ X∗∗ be an arbitraryvirtual E-class. Use the balance of the pair 〈Q, τ〉 to show that there must bea unique virtual F -class d ∈ Y ∗∗ such that R × Q 〈x, y〉 ∈ Agen for somerealizations x, y of the virtual classes c, d. Use the balance of the pair 〈Q, τ〉again to show that for every virtual F -class d ∈ Y ∗∗, there either must be avirtual E-class c ∈ X∗∗ such that R × Q 〈x, y〉 ∈ Agen for some realizationsx, y of the virtual classes c, d, or it must be the case that R × Q y ∈ τ forsome realization y of the virtual class d.

To see that 〈R, τg〉 is equivalent to 〈Q, τ〉, strengthen τ if necessary so it is aname for an actual element of P and notice that in the R×Q extension, τ ∈ τgholds. The equivalence then follows from Proposition 5.2.4.

Finally, (3) is obvious.

Corollary 6.6.4. Let E,F be Borel equivalence relations on respective Polishspaces X,Y with uncountably many classes. The E,F -collapse poset is balancedif and only if λ(E) ≤ λ(F ).

The collapse posets exemplify an important phenomenon: a σ-closed Suslin forc-ing whose balanced status cannot be decided in ZFC. There are Borel equiva-lence relations E, F for which the status of the inequality λ(E) ≤ λ(F ) cannotbe decided in ZFC (as can be seen for example from the combination of Corol-laries 2.5.18 and 2.5.15) and for them the balanced status of the correspondingcollapse forcing is undecidable as well.

Definition 6.6.5. Let E ⊂ F be Borel equivalence relations on a Polish spaceX. Let K = KE,F be the simplicial complex of all finite sets a ⊂ X such thatfor x0, x1 ∈ X, x0 E x1 is equivalent to x0 F x1. We will refer to the associatedposet PK as the E,F -transversal poset.

It is immediate that the simplicial complex KE,F is a Borel E-quotient complex.The terminology comes from the examples below; the posets of this type areused to uniformize various relations in products of quotient spaces. The firstorder of business is to classify the balanced conditions. Call a total functionf from the F -quotient space to the virtual E-quotient space a virtual selectionfunction if for every F -class c the functional value f(c) is a virtual E-class whichis (forced to be) a subset of c. For a virtual selection function f let τf be theColl(ω,iω1)-name for the set of all conditions p in the uniformization poset suchthat for each class c represented in the ground model, p contains some pointx ∈ X which is a realization of the virtual E-class f(c). It is not difficult to seethat the pair 〈Coll(ω,iω1

), τf 〉 is a P -pin.


Theorem 6.6.6. Let E ⊂ F be Borel equivalence relations on a Polish spaceX and suppose that F is pinned. Let P be the E,F -transversal poset. Then

1. for every virtual selection function f , the pair 〈Coll(ω,iω1), τf 〉 is bal-

anced;

2. for every balanced pair 〈Q, τ〉 there is a virtual selection function f suchthat the pairs 〈Coll(ω,iω1), τf 〉 and 〈Q, τ〉 are equivalent;

3. distinct virtual selection functions yield nonequivalent balanced pairs.


Proof. To see (1), let V [H0] and V [H1] be mutually generic extensions of theground model and p0 ∈ V [H0] and p1 ∈ V [H1] be conditions in the set τf/H0 =τf/H1; we have to show that p0, p1 ∈ P are compatible. To this end, supposethat x0 ∈ p0 and x1 ∈ p1 are points; we must show that if x0 F x1 then x0 E x1

holds. A mutual genericity argument shows that x0, x1 are realizations of thesame virtual F -class c. By the definition of the name τf , it must be the case thatx0, x1 are both realizations of the virtual E-class f(c) and therefore E-related.

For (2), suppose that 〈Q, τ〉 is a balanced pair. To find the total functionf , let c be an F -class. There must be a virtual E-class f(c) such that Q for some point x ∈ τ ∩ c x is a realization of f(c); otherwise, one could findconditions q0, q1 ∈ Q and names for strengthenings τ0, τ1 of τ and names x0, x1

for elements of c such that 〈q0, q1〉 Q×Q ¬(x0)left F (x1)right. This wouldviolate the balance of the pair 〈Q, τ〉 as 〈q0, q1〉 τ0, τ1 ∈ P are incompatibleconditions. Let f : c 7→ dc be the resulting function.

To see that 〈R, τf 〉 is a balanced pair equivalent to 〈Q, τ〉, strengthen τ ifnecessary to ensure that τ is a name for an actual element of P , and observethat Q×Coll(ω,iω1

) τ ∈ τf . The equivalence of the two balanced pairs thenfollows from Proposition 5.2.4.

Finally, (3) is obvious. The balance of the poset P is now a trivial applicationof the Axiom of Choice: every condition p ∈ P can be extended to a maximalset A ⊂ X such that E � A = F � A. Let f be the function which to eachF -class c assigns the E-class c ∩A, and observe that the pair 〈Coll(ω,iω1), τf 〉is balanced by (1) and is below p.

Corollary 6.6.7. Let E ⊂ F be Borel equivalence relations on a Polish spaceX such that F is pinned. Let P be the E,F -uniformization poset. Then P isbalanced.

Example 6.6.8. A poset introducing a transversal to a pinned Borel equiva-lence relation F on a Polish space X. Just let E be the identity on the spaceX and consider the E,F -transversal poset. The generic set is an F -transversal.By Theorem 6.6.6, the balanced conditions are classified by F -transversals.

Example 6.6.9. A poset introducing a complete countable section to a pinnedBorel equivalence relation G on a Polish space Y . Let X = Y ω and let B ⊂ X


be the Borel set of all elements x ∈ X whose range consists of pairwise G-relatedpoints. Let F be the Borel equivalence relation on X connecting points x0, x1

if either they are equal or else they are both in B and their ranges are subsetsof the same G-class. Let E = F2 ∩ F . It is not difficult to check that E ⊂ Fare Borel equivalence relations on X and F is pinned. Consider the associatedE,F -transversal poset. The generic set is a countable complete section of therelation E. In view of the fact that virtual F2-classes are classified by nonemptysubsets of Y , Theorem 6.6.6 says that the balanced conditions are classified byarbitrary sets A ⊂ Y which have nonempty intersection with every G-class.

Numerous examples of uniformization posets arise in the context of countableBorel equivalence relations. If E is a countable Borel equivalence relation ona Polish space Y and P is a poset adding a choice of a structure of some typeon each E-class with countable approximations, then P can be represented as atransversal poset in the sense of Definition 6.6.5. The following example spellsout the details.

Example 6.6.10. Let G be a countable Borel equivalence relation on a Polishspace Y with all equivalence classes infinite. Let P be the poset of all count-able functions on the G-quotient space assigning to each equivalence class aZ-ordering on it; the poset P serves to introduce a (discontinuous) Z-action onY such that G is its orbit equivalence relation. It is not difficult to cast P as atransversal poset in the sense of Definition 6.6.5. Let X = (Y 2)ω and let B ⊂ Xbe the Borel set of all elements x ∈ X which enumerate a linear ordering of awhole single G-class of ordertype Z. Let F be the equivalence relations on Xconnecting points x0, x1 ∈ X if either they are equal or else they both belongto B and enumerate a linear ordering of the same G-class. Let E = F2 ∩ F . Itis not difficult to check that both E,F are pinned Borel equivalence relations.Consider the associated E,F -transversal poset. The generic set selects one Z-type ordering for each G-class. Theorem 6.6.6 says that the balanced conditionsare classified by all functions f assigning each G-class a linear ordering of itselements isomorphic to Z.

Example 6.6.11. Let X = ((2ω)ω)2 and consider the Borel set B ⊂ X con-sisting of all pairs 〈y0, y1〉 ∈ X such that rng(x0) ∩ rng(x1) = 0. Let F bethe equivalence relation connecting points 〈y0, y1〉 and 〈z0, z1〉 if they are eitherequal or both in B and y0 F2 z0. Let E be the equivalence relation F2 × F2

intersected with F . The resulting E,F -transversal poset is designed to add afunction assigning to each nonempty countable set a single nonempty countableset disjoint from it. However, the equivalence relation F is not pinned and soTheorem 6.6.6 does not apply to show that the uniformization poset is balanced.Indeed, in ZF one can prove that existence of a maximal KE,F -set implies theexistence of an ω1-sequence of pairwise distinct E-classes. To see this, let A ⊂ Xbe such a maximal set, suppose c0 ⊂ 2ω is an arbitrary nonempty countable set,and by transfinite recursion on α ∈ ω1 define cα as the unique nonempty count-able set such that for some enumerations y0 of

⋃β∈α cβ and y1 of cα it is the

case that 〈y0, y1 ∈ B. It is clear that the sets cα for α ∈ ω1 are nonempty and


pairwise disjoint and therefore distinct. Finally, note (Theorem 9.1.1) that inbalanced extensions of the symmetric Solovay model there are no uncountablesequences of distinct equivalence classes of Borel equivalence relations such asF2.

If the larger equivalence relation F in Definition 6.6.5 is not pinned, the circum-stances must be quite peculiar for the E,F -transversal poset to be balanced.One class of examples of this type is captured by the following theorem. ForBorel equivalence relations E ⊂ F on a Polish space X write X∗∗E , X∗∗F to betheir respective virtual quotient spaces. A function f : X∗∗F → X∗∗E is a virtualE,F -selection function if for every virtual F -class c, the virtual E-class f(c)is forced to be a subset of c. For each E,F -selection function f let τf be theColl(ω,iω1

)-name for the set of all countable subsets a ⊂ X such that for eachvirtual F -class c, a contains a realization of the virtual E-class f(c).

Theorem 6.6.12. Let E ⊂ F be Borel equivalence relations on a Polish spaceX such that each F -class consists of countably many E-classes. Let P be theE,F -transversal poset. Then

1. for every virtual E,F -selection function f : X∗∗F → X∗∗E , the pair 〈Coll(ω,iω1), τf 〉

is balanced;

2. for every balanced pair 〈Q, τ〉 there is a virtual selection function f suchthat the balanced pairs 〈Q, τ〉, 〈Coll(ω,iω1

), τf 〉 are equivalent;

3. distinct virtual selection functions yield nonequivalent balanced pairs.


Proof. The argument hinges on a simple claim of independent merit:

Claim 6.6.13. Let V [H] be a generic extension of V , and in V [H] let c be anF -class which is a realization of a ground model virtual F -class. Then c is aunion of realizations of ground model virtual E-classes.

Proof. Suppose towards a contradiction that this fails. Then in V there has to bea poset P and a P -name τ for an element of the space X such that τ is F -pinned,but below no condition of P , τ is E-pinned. Let M be a countable elementarysubmodel of a large structure containing E,F, P, τ and use Proposition 1.7.10 tofind a perfect collection {gy : y ∈ 2ω} of filters on P ∩M mutually generic overM . Let xy = τ/gy ∈ X. For distinct binary sequences y, z ∈ 2ω, M [gy, gz] |=xy F xz and ¬xy E xz by the forcing theorem applied in M . By the Mostowskiabsoluteness for the model M [gy, gz], it follows that xy F xz and ¬xy E xzholds in V . However, this means that the unique F -class of the points xy fory ∈ 2ω contains perfectly many pairwise E-unrelated points, contradicting theinitial assumptions on the equivalence relations E,F .

Now, for (1) suppose that V [H0], V [H1] are mutually generic extensions con-taining respective conditions p0, p1 below τf ; we must show that p0, p1 are com-patible. This is to say, p0 ∪ p1 is a KE,F -set; in other words, if x0 ∈ p0 and


x1 ∈ p1 are F -related points then they are in fact E-related. To see this, bya mutual genericity argument x0, x1 are realizations of some virtual F -class c;by the definition of the virtual condition τf , they must be realizations of thevirtual E-class f(c) and therefore must be F -related.

For (2), first use a balance argument and Claim 6.6.13 to argue that foreach virtual F -class c there must be a unique virtual E-class d which is forcedto be a subclass of c and such that Q τ contains a realization of d. Letf : X∗∗F → X∗∗E be the function recording the correspondence c 7→ d and observethat Q × Coll(ω,iω1

) τ ≤ τf holds. (2) then follows by a reference toProposition 5.2.4.

Finally, (3) is obvious. The balance of the poset P follows from the axiomof choice: every condition p ∈ P can be extended to a virtual selection functionby virtua of Claim 6.6.13.

Example 6.6.14. Let Γ be a Borel graph on a Polish space X which containsno odd length cycles such that the G-path connectedness equivalence relationF on X is Borel. We wish to add a 2-coloring of the graph Γ. To this end,let E be the Borel equivalence relation on X relating x to y if x E y holdsand some (equivalently, every) Γ-path from x to y is of even length. It is clearthat every F -class consists of exactly two E-classes. The E,F -transversal posetselects exactly one E-class from each F -class and therefore adds a bipartizationof the graph Γ. The balanced conditions are classified by functions which selectexactly one virtual E-class from each virtual F -class.

Example 6.6.15. Let F be the F2-equivalence relation on the space X = (2ω)ω.Let E be the equivalence relation on X connecting points x0 and x1 if rng(x0) =rng(x1) and x0(0) = x1(0). Clearly, every F -class consists of only countablymany E-classes. An E,F -transversal can be viewed as a function which selectsa single element from each nonempty countable subset of 2ω. It is instructiveto inspect the balanced virtual conditions provided by Theorem 6.6.12. Aninspection reveals that a balanced virtual condition is a function which to eachnonempty subset of X assigns one of its elements, i.e. a selection function onP(X). In consequence, the existence of balanced conditions for P is in ZFequivalent to the statement that the space 2ω can be well-ordered.

Chapter 7

Ultrafilter forcings

Many applications of the axiom of choice rely on nonprincipal ultrafilters onω and their combinatorial properties. In this section, we will show that inseveral cases, natural attempts to add an ultrafilter result in balanced forcingsand moreover the resulting generic ultrafilter can be characterized in a simpleway. The following definition and proposition will play a central role in severalarguments.

Definition 7.0.1. Let U be a nonprincipal ultrafilter on a set dom(U). Aninfinite set a ⊂ dom(U) in some forcing extension diagonalizes U if for everyb ∈ U , a \ b is finite.

Proposition 7.0.2. Let U be a nonprincipal ultrafilter on dom(U). Supposethat Q0, Q1 are posets and τ0, τ1 are respective names for infinite subsets ofdom(U) which diagonalize U . Then Q0 ×Q1 τ0 ∩ τ1 is infinite.

Proof. The most appealing argument uses the general Proposition 1.7.8 on prod-uct forcing extensions. Let H0, H1 ⊂ Q0, Q1 be mutually generic filters and leta0 = τ0/H0 and a1 = τ1/H1. Suppose towards a contradiction that the inter-section a0 ∩ a1 is finite. By Proposition 1.7.8, there are sets b0, b1 ⊂ dom(U) inthe ground model such that a0 ⊂ b0, a1 ⊂ b1, and b0 ∩ b1 is finite. At most oneof the sets b0, b1 belongs to the ultrafilter U . If, say, b0 /∈ U , then ω \ b0 ∈ U ,violating the assumption that a0 diagonalizes the ultrafilter U .

7.1 A Ramsey ultrafilter

The most elementary attempt at adding a nonprincipal ultrafilter is the poset Pof infinite subsets of ω ordered by inclusion. Here, the classification of balancedconditions is particularly appealing and simple.

Definition 7.1.1. Let U be a nonprincipal ultrafilter on ω. The symbol τUdenotes the Coll(ω,U)-name for the analytic collection of all infinite sets a ⊂ ωwhich diagonalize the ultrafilter U .

157

158 CHAPTER 7. ULTRAFILTER FORCINGS

Theorem 7.1.2. Let P be the partial order of infinite subsets of ω, ordered byinclusion.

1. The pair 〈Coll(ω,U), τU 〉 is a balanced virtual condition for P for everynonprincipal ultrafilter U on ω;

2. If 〈Q, τ〉 is a balanced pair for P , then there is a nonprincipal ultrafilterU such that 〈Q, τ〉 is equivalent to 〈Coll(ω,U), τU 〉;

3. distinct nonprincipal ultrafilters yield inequivalent balanced virtual condi-tions.


Proof. For (1), note that the pair in fact is a virtual condition: in Coll(ω,U),the set U is countable and so τU is an analytic subset of P , and the evaluationof τU does not depend on the particular generic filter on the collapse poset. Thebalance of the pair follows immediately from Proposition 7.0.2 applied to U .

For (2), first use the balance of the name τ to note that for every set a ⊂ ω,it must be the case that either Q τ ⊂ a up to a finite set, or Q τ ∩ a isfinite. Let U be the set of all a ⊂ ω for which the first option occurs. It isimmediate that U is a nonprincipal ultrafilter on ω. The equivalence of 〈Q, τ〉with 〈Coll(ω,U), τU 〉 is immediately clear from Proposition 7.0.2.

For (3), not that if U0, U1 are distinct ultrafilters on ω then there is a seta ⊂ ω such that a ∈ U0 and ω \ a ∈ U1. Thus the balanced names τU0 and τU1

represent incompatible virtual conditions as the former is below a and the otheris below ω \ a.

To prove the last sentence of the theorem, for any infinite set a ⊂ ω there isa nonprincipal ultrafilter U containing a as an element. A reference to item (1)then completes the argument.

7.2 Fubini powers of the Frechet ideal

The most natural attempt to force an ultrafilter which is not a P-point with aσ-closed subset is encapsulated in the following definition.

Definition 7.2.1. For the duration of this section, write I to be the ideal onω × ω which is the Fubini product of the Frechet ideal on ω with itself. Thatis, a set a ⊂ ω × ω belongs to the ideal I just in case a has only finitely manyinfinite vertical sections. The Fin×Fin poset P isthe partial order of I-positivesets, ordered by q ≤ p if q \ p ∈ I.

It is not difficult to see that P is a σ-closed forcing; the sets in the genericfilter on P form an ultrafilter disjoint from I. This ultrafilter, by virtue of thepartition of the domain set ω × ω into the I-small vertical sections such thatevery transversal is again in I, is not a P-point; its rich combinatorial propertieswere investigated for example in [11, 89].

7.2. FUBINI POWERS OF THE FRECHET IDEAL 159

It turns out that the poset P is balanced, and the balanced virtual conditionsallow a simple classification. Let ω∗ be the set of all nonprincipal ultrafilters onω. Let W be an ultrafilter on the set ω × ω∗ which contains no set n × ω∗ forany number n ∈ ω. Write τW for the name on Coll(ω,P(ω × ω∗)) for the setof all conditions p ∈ P such that there is a set {〈ni, Ui〉 : i ∈ ω} ⊂ (ω × ω∗)Vdiagonalizing the ultrafilter W , the numbers ni for i ∈ ω are pairwise distinct,and the set p ⊂ ω × ω has the following property: only the vertical sectionspni for some i ∈ ω are nonempty, and each vertical section pni is infinite anddiagonalizes the ultrafilter Ui.

Theorem 7.2.2. Let P be the Fin×Fin poset.

1. For every ultrafilter W on the set ω × ω∗ containing no set n × ω∗ forn ∈ ω, the pair 〈Coll(ω,P(ω × ω∗)), τW 〉 is balanced in P ;

2. for every balanced pair 〈Q, σ〉 there is an ultrafilter W on the set ω × ω∗,containing no set n × ω∗ for n ∈ ω, such that the balanced pairs 〈Q, σ〉and 〈Coll(ω,P(ω × ω∗)), τW 〉 are equivalent;

3. distinct ultrafilters give rise to inequivalent balanced conditions.


Proof. Towards (1), suppose that V [H0], V [H1] are mutually generic exten-sions containing respective sets p0, p1 ⊂ ω × ω such that there are sets a0 ={〈n0

i , U0i 〉 : i ∈ ω} and a1 = {〈n1

i , U1i 〉 : i ∈ ω} in the respective models which

diagonalize the ultrafilter W , and the vertical sections (p0)n0i

diagonalize the

ultrafilter U0i for all i ∈ ω, and the vertical sections (p1)n1

idiagonalize the ultra-

filter U1i for all i ∈ ω as well. It will be enough to show that p0, p1 are compatible

in the poset P ; i.e. the set p0 ∩ p1 has infinitely many infinite vertical sections.To this end, use Proposition 7.0.2 for the sets a0, a1 and the ultrafilter W

to see that a0 ∩ a1 is infinite. This means, that the set b = {n ∈ ω : ∃U 〈n,U〉 ∈a0 ∩ a1} is infinite. For each number n ∈ b, the vertical sections (p0)n and(p1)n diagonalize the same ultrafilter U on ω for which 〈n,U〉 ∈ a0 ∩ a1 holds.By another application of Proposition 7.0.2, the set (p0)n ∩ (p1)n is infinite. Itfollows that p0 ∩ p1 /∈ I as desired.

Towards (2), strengthen σ if necessary so that σ is a name for an actual singlecondition in the poset P as opposed to an analytic set of conditions. ReplacingQ with a poset collapsing the cardinality of P(ω)V to ℵ0 and strengthening thename σ if necessary we may assume that Q forces that for every n ∈ ω, if thevertical section σn is infinite, then for every set a ⊂ ω in the ground model,either σn ⊂ a or σn ∩ a = 0 modulo finite. Let Un be the name for the set ofall sets a ⊂ ω in the ground model for which the former alternative prevails;define Un = 0 if the vertical section σn is finite. Passing to a condition in Q andstrengthening σ again, we may assume that either Q forces that for all n ∈ ωsuch that σn is infinite Un /∈ V holds, or Q forces that for all n ∈ ω such thatσn is infinite Un ∈ V holds.


Now, the former alternative is impossible as it contradicts the balance ofthe pair 〈Q, τ〉. To see this, let H0, H1 ⊂ Q be mutually generic filters andlet p0 = σ/H0 and p1 = σ/H1. To reach the contradiction, we will show thatall vertical sections of p0 ∩ p1 are finite. Suppose not, and let n ∈ ω be suchthat (p0)n ∩ (p1)n is infinite. Then Un/H0 = Un/H1, which by the productforcing theorem implies that Un/H0 ∈ V and violates the former alternativeassumption.

Thus, the latter alternative prevails, and the set η = {〈n, Un〉 : σn is infinite}is forced to be a subset of (ω × ω∗)V . Strengthening σ again, we may assumethat Q forces that for every set b ⊂ ω×ω∗ in the ground model, either η ⊂ b orη ∩ b = 0 modulo finite.

Claim 7.2.3. For every set b ⊂ ω × ω∗, either Q η ⊂ b modulo finite, orQ η ∩ b is finite.

Proof. Suppose towards a contradiction that the conclusion fails for some setb ⊂ ω × ω∗. Then, there must be conditions q0, q1 ∈ Q forcing the formerand latter alternative respectively. Let H0, H1 ⊂ Q be mutually generic filterscontaining the conditions q0, q1 respectively. Write p0 = σ/H0 and p1 = σ1/H1;we will reach a contradiction by showing that all but finitely many verticalsections of the set p0 ∩ p1 are finite, violating the balance assumption on σ.

Write c0 = η/H0 and c1 = η/H1. By the contradictory assumption, theintersection c0 ∩ c1 is finite. Let n ∈ ω be a natural number which is not in thefinite projection of the set c0 ∩ c1 to ω; we will show that (p0 ∩ p1)n is finite.Either (p0)n or (p1)n are finite sets, in which case we are done, or both (p0)nand (p1)n are infinite. In the latter case, there are ultrafilters U0, U1 on ω in theground model such that 〈n,U0〉 ∈ c0 and 〈n,U1〉 ∈ c1. Since the number n is notin the projection of c0 ∩ c1 to ω, the ultrafilters U0, U1 must be distinct. Since(p0)n diagonalizes U0 and (p1)n diagonalizes U1, the intersection (p0)n ∩ (p1)nmust be finite in this case as well.

Let W be the set of all subsets of b ⊂ ω × ω∗ for which Q η ⊂ b modulofinite; by the claim, this is an ultrafilter on ω × ω∗. The definitions show thatColl(ω,P(ω × ω∗)) × Q σ ∈ τF . Then, Proposition 5.2.4 shows that thebalanced pairs 〈Q, σ〉 and 〈Coll(ω,P(ω × ω∗)) are equivalent as desired.

Item (3) is immediate. For the last sentence, suppose that p ∈ P is acondition. Let a = {n ∈ ω : pn is infinite}; the set a is infinite. For each n ∈ ωlet Un be a nonprincipal ultrafilter on ω containing pn as an element. Let W bea nonprincipal ultrafilter on the set ω × ω∗ containing the set {〈n,Un〉 : n ∈ a}.It is not difficult to see that the balanced pair 〈Coll(ω,P(ω×ω∗)), τW 〉 is belowthe condition p, proving the balance of the poset P .

7.3 Ramsey sequences of structures

There is a family of forcings present in several papers concerning the Rudin–Keisler order on ultrafilters with strong Ramsey-type properties [23]. The family

7.3. RAMSEY SEQUENCES OF STRUCTURES 161

is parametrized by sequences of structures tied by a partition property as in thefollowing definition.

Definition 7.3.1. A Ramsey sequence of finite structures is a sequence A =〈An : n ∈ ω〉 of finite structures in the same language, with pairwise disjointdomains, such that

1. for every natural number n ∈ ω, An+1 contains an isomorphic copy of An;

2. for every k < n ∈ ω there is m > n such that the structural Ramseyproperty Am → (An)Ak2 holds.

For each Ramsey sequence A = 〈An : n ∈ ω〉 of finite structures, we will use thefollowing notation. dom(A) stands for

⋃n dom(An). Given a number n ∈ ω,

the symbol Dn stands for the set of all finite sets d such that for some m ∈ ω,d ⊂ dom(Am) and Am � d is isomorphic to An. For every set p ⊂ dom(A), thesymbol pAn stands for P(p) ∩ Dn. Each Ramsey sequence of finite structureshas a natural σ-closed poset associated to it.

Definition 7.3.2. Let A = 〈An : n ∈ ω〉 be a Ramsey sequence of finite struc-tures. Let PA denote the poset of all sets p ⊂ dom(A) such that pAn is nonemptyfor each n ∈ ω. The ordering is defined by q ≤ p if q ⊆ p modulo finite.

It is not difficult to use the Ramsey property of the sequence A to see thata generic filter on PA is in fact an ultrafilter on dom(A); the combinatorialproperties of this generic ultrafilter are the reason for the study of these posetsin [8]. The posets of the form PA are balanced. The balanced virtual conditionsare parametrized by sequences of ultrafilters as opposed to single ultrafilters inthe case of P(ω) modulo finite.

Definition 7.3.3. Let A = 〈An : n ∈ ω〉 be a Ramsey sequence of finite struc-tures. An A-sequence of filters is a sequence 〈Fn : n ∈ ω〉 such that

1. for every number n ∈ ω, Fn is a filter on the set Dn;

2. for all numbers n,m ∈ ω and every selection 〈ai : i ∈ n〉 of sets in therespective filters Fi for i ∈ n, there is a set d ∈ Dm such that dAi ⊂ aiholds for all i ∈ n.

Whenever F = 〈Fn : n ∈ ω〉 is anA-sequence of ultrafilters, define the Coll(ω, 2ω)-name τF for the set of all conditions p ∈ P such that for every n ∈ ω and everyset a ∈ Fn, pAn ⊂ a modulo finite.

Theorem 7.3.4. Let A = 〈An : n ∈ ω〉 be a Ramsey sequence of finite struc-tures.

1. For every A-sequence F of ultrafilters, the pair 〈Coll(ω, 2ω), τF 〉 is balancedin the poset PA;


2. for every balanced pair 〈Q, σ〉 there is a A-sequence F of ultrafilters suchthat the balanced pairs 〈Q, σ〉 and 〈Coll(ω, 2ω), τF 〉 are equivalent;

3. distinct A-sequences of ultrafilters yield inequivalent balanced pairs.

In particular, the poset PA is balanced.

Proof. Write P = PA. For the first item, it is necessary to argue that infact Coll(ω, τF ) τF 6= 0. To see this, work in the Coll(ω, 2ω) extension,for each number n ∈ ω let Dn = dom(Fn) and fix an enumeration 〈ain : i ∈ ω〉of all elements of Fn. By induction on n ∈ ω, repeatedly using item (2) ofDefinition 7.3.3 build sets dn ∈ Dn such that for every i ∈ n the set dAin isa subset of

⋂j∈n a

ji . It is immediate that the condition p =

⋃n dn ∈ P is an

element of τF .For the balance, suppose that R0, R1 are posets and σ0, σ1 are their respec-

tive names for conditions in P such that R0 forces that for every n ∈ ω andevery set a ∈ Fn, σAn0 ⊂ a modulo finite holds, and similarly for R1, σ1. Wehave to show that R0 × R1 forces the conditions σ0, σ1 to be compatible in P ,which is the same as to say that R0 × R1 forces the intersection σ0 ∩ σ1 tocontain a copy of An for every number n ∈ ω. To this end, note that the setsσAn0 and σAn1 are forced to diagonalize the ultrafilter Fn. By Proposition 7.0.2,their intersection is forced to be infinite, in particular nonempty.

For the second item, we use a simple claim.

Claim 7.3.5. Let p ∈ P be a condition, n ∈ ω be a number, and let a ⊂ Dn bea set. Then there is q ≤ p such that either qAn ⊂ a or qAn ∩ a = 0.

Proof. For each k ∈ ω find a number mk ∈ ω such that Amk → (Ak)An2 . Findpairwise disjoint finite sets dk ⊂ p in Dmk for each k ∈ ω. Use the partitionproperty to find sets d′k ⊂ dk such that d′k ∈ Dk and (d′k)An is either a subset ofa or disjoint from a. One of the options prevails for infinitely many numbers k.For definiteness, assume that the set b = {k ∈ ω : (d′k)An ⊂ a} is infinite. Letq =

⋃k∈b d

′k and observe that q ≤ p works.

Now let 〈Q, σ〉 be a balanced pair. Replacing Q with Q × Coll(ω, 2ω) andrepeatedly strengthening p using the claim, we may assume that for every n ∈ ωand every set a ⊂ Dn, σAn is forced to be either a subset of a or disjointfrom a modulo finite. By a balance argument, it must be the case that eitherQ σAn ⊂ a modulo finite, or Q σAn ∩ a = 0 modulo finite. Let Fn be thecollection of all sets a ⊂ Dn for which the former option prevails. It is clearthat Fn is an ultrafilter and that F = 〈Fn : n ∈ ω〉 is an A-sequence. SinceQ×Coll(ω, 2ω) σ ∈ τF , by the first item and Proposition 5.2.4 it must be thecase that the balanced pairs 〈Q, σ〉 and 〈Coll(ω, 2ω), τF 〉 are equivalent balancedpairs.

The third item is immediate. The last sentence of the theorem is not anentirely formal consequence of the previous work. It needs the following appli-cation of the axiom of choice.

7.3. RAMSEY SEQUENCES OF STRUCTURES 163

Claim 7.3.6. Every A-sequence of filters can be extended to a A-sequence ofultrafilters.

Proof. Order the A-sequences of filters by coordinatewise inclusion: F ≤ G if forevery n ∈ ω, Fn ⊆ Gn holds. It is immediate from the definitions that every ≤-chain has an upper bound–the coordinatewise union of the A-sequences of filtersin the chain. By an application of Kuratowski–Zorn lemma, every A-sequenceof filters can be extended into a ≤-maximal one. It only remains to prove thatany ≤-maximal A-sequence of filters is in fact a sequence of ultrafilters.

Let F be a ≤-maximal A-sequence of filters, let n ∈ ω be a number, andlet b ⊂ Dn be a set; we claim that either b or its complement belongs to Fn.Suppose towards a contradiction that neither is the case. By the maximalityof F it must be the case that the sequences G0, G1 of filters obtained from Fby adding b (or the complement of b, respectively) to the filter Fn are not A-sequences. This must be witnessed by some sets 〈a0

i : i ∈ k〉 and 〈a1i : i ∈ k〉 for

some number k ∈ ω respectively, and some number m ∈ ω: these are sets inFi such that there is no d ∈ Dm such that for all i ∈ k, i 6= n dAi ⊂ a0

i anddAn ⊂ a0

n ∩ b, and there is no d ∈ Dm such that for all i ∈ k, i 6= n dAi ⊂ a1i

and dAn ⊂ a1n \ b. Since F is a sequence of filters, for all i ∈ k a0

i ∩ a1i ∈ Fi

holds. Let m′ ∈ ω be a number such that Am′ → (Am)An holds. Since F is aA-sequence, there is d′ ∈ Dm′ such that for all i ∈ k, (d′)Ai ⊂ a0

i ∩a1i holds. Use

the partition assumption to find a a set d ⊂ d′ in Dm such that either dAn ⊂ bor dAn ∩ b = 0. The condition q violates the choice of either the sets 〈ai0 : i ∈ k〉or the sets 〈ai1 : i ∈ k〉.

Now, suppose that p ∈ P is a condition. Consider the sequence F of filters Fngenerated by sets cofinite in pAn . Clearly, F is an A-sequence. By the claim, Fcan be extended to a A-sequence G of ultrafilters. Then 〈Coll(ω, 2ω), τG〉 is abalanced pair which is easily seen to be below p as required.

For every Ramsey sequence A = 〈An : n ∈ ω〉, the poset PA introduces anultrafilter on the set D0 as the set of all subsets a ⊂ D0 such that for somecondition p ∈ P in the generic ultrafilter, pA0 ⊂ a. Claim 7.3.5 together with anobvious density argument show that this formula indeed defines an ultrafilter Uon D0, which will be referred to as the PA-generic ultrafilter. It is clear that thegeneric filter on PA can be reconstructed from U , and therefore the potential forconfusion is minimal. The combinatorial and Rudin–Keisler properties of thegeneric ultrafilter have been the main reason for the study of the partial ordersof the type PA [8]. It is not difficult to see that it is a P-point.

Example 7.3.7. Let A be the sequence 〈An : n ∈ ω〉 where An is the linearorder on n many elements. The Ramsey theorem implies that this is in fact aRamsey sequence of structures. The PA-generic ultrafilter is a P-point which isweakly Ramsey and not a Q-point [8, Theorem 4.9].

Example 7.3.8. Let k ∈ ω and let A be a sequence enumerating all orderedfinite graphs which contain no clique of size k. The fact that this is a Ramsey


sequence of structures follows from [72]. The PA-generic ultrafilter is a P-pointU such that U → (U, k)2 but not U → (U, k + 1)2 [8, Theorem 4.11]

7.4 Semigroup ultrafilters

In this section, we show that in a natural forcing connected with a countablesemigroup, the balanced conditions exist and are classified by idempotent ul-trafilters. This connects the theory of balanced forcing extensions with Ramseytheory and dynamics.

Definition 7.4.1. Let Γ be a countable semigroup. The poset PΓ is definedas follows. The conditions of PΓ are elements of Γω. The ordering is definedby q ≤ p if there are nonempty finite sets an ⊂ ω for all n ∈ ω such thatmax(an) < min(an+1) and Πm∈anp(m) = q(n), where the products are alwaystaken in the increasing order.

It is not difficult to see that given a condition in PΓ, shifting its entries to the leftand/or changing finitely many entries does not change the separative quotientequivalence class of the condition. It follows that the separative quotient of PΓ

is a σ-closed poset. The purpose of the poset PΓ is clear from the followingdefinition and proposition.

Definition 7.4.2. Let A be a collection of subsets of Γ. A sequence p ∈ PΓ

diagonalizes A if for every set b ∈ A there is n ∈ ω such that for every nonemptyfinite set a ⊂ ω\n, Πm∈ap(m) ∈ b holds. The sequence p ∈ PΓ sorts out A if forevery set b ∈ A there is n ∈ ω such that for every nonempty finite set a ⊂ ω \n,Πm∈ap(m) ∈ b holds, or for every nonempty finite set a ⊂ ω \ n, Πm∈ap(m) /∈ bholds. If b ∈ A and the former alternative occurs, we say that p accepts b, if thelatter alternative occurs then p declines b.

Proposition 7.4.3. For every condition p ∈ PΓ and every countable set A ⊂P(Γ), there is a condition q ≤ p which sorts out A.

Proof. Let A = {bn : n ∈ ω}. By induction on n ∈ ω build a descending sequencepn of conditions in PΓ such that p = p0 and for all n ∈ ω, for all nonempty finitesets a ⊂ ω it is the case that

∏m∈a pn+1(m) ∈ bn holds, or for all nonempty

finite sets a ⊂ ω it is the case that∏m∈a pn+1(m) ∈ bn holds. To perform the

induction step, use Hindman’s theorem on the partition πn of nonempty finitesubsets of ω defined by πn(a) = 1 if

∏m∈a pn(m) ∈ bn holds.

In the end, let q ∈ PΓ be defined by q(n) = pn(n) and observe that theconclusion of the proposition is satisfied.

Suppose that G ⊂ PΓ is a generic filter. By Proposition 7.4.3 and a densityargument, the set U ⊂ P(Γ) of all sets b ⊂ Γ such that some condition p ∈ Gaccepts b, is an ultrafilter. Also, G can be recovered from U as the set of allconditions p ∈ PΓ such that for all n ∈ ω, the set {

∏m∈a p(m) : a ⊂ ω is a

nonempty finite set and min(a) > n} belongs to the ultrafilter U .

7.4. SEMIGROUP ULTRAFILTERS 165

It turns out that the balanced conditions in the poset PΓ correspond to awell-known concept from topological dynamics and Ramsey theory.

Definition 7.4.4. [40, Section 4.1] Let 〈Γ, ·〉 be a countable semigroup.

1. For ultrafilters U0, U1 on Γ, let U0 · U1 = {A ⊂ Γ: {γ ∈ Γ: {δ ∈ Γ: γ · δ ∈A} ∈ U1} ∈ U0};

2. an ultrafilter U on Γ is an idempotent if U · U = U .

It turns out that · is a semi-continuous operation on the space βΓ of all ultra-filters on Γ. The fundamental Ellis–Numakura theorem [40, Section 2.4] saysthat every closed subsemigroup of βΓ contains an idempotent ultrafilter. Theidempotent ultrafilters have an alternative diagonalization description.

Proposition 7.4.5. Let U be an ultrafilter on Γ. The following are equivalent:

1. in Coll(ω,P(Γ)) extension there is a condition in PΓ diagonalizing U ;

2. for every countable subset of U , there is a condition in PΓ diagonalizingit;

3. U is an idempotent.

Proof. For the implication (1)→(2), if U ′ ⊂ U is a countable set, then in someforcing extension there is a condition p ∈ P which diagonalizes U and thereforeU ′. By a Mostowski absoluteness argument between the ground model and theforcing extension, there must be a condition p ∈ P diagonalizing U ′.

For the implication (2)→(1), consider the partial order Q = [P(U)]ℵ0 mod-ulo the nonstationary ideal. Let G ⊂ Q be a filter generic over V and letj : V →M be the generic ultrapower. Then M is an ω-model which is possiblyillfounded. Moreover, M contains the set j′′U as the equivalence class of theidentity function on [P(U)]ℵ0 , and M |= j′′U ⊂ U is a countable set by the Lostheorem. Thus, if the ground model satisfies (2), then by elementarity M (andso V [G]) contains a condition p ∈ P which diagonalizes j′′U and therefore U .The existence of such a condition then transfers to any Coll(ω,P(Γ)) extensionof the ground model. This confirms (1).

(3) implies (2): this is the content of Galvin–Glazer theorem or its folkloricvariation for countable sets, [40, Chapter 5]. Finally, the negation of (3) impliesthe negation of (2). To see this, if U is not an idempotent, there must be a setA ∈ U such that A /∈ U ·U . This means that the set B = {γ ∈ Γ: {δ ∈ Γ: γ ·δ /∈A} ∈ U} belongs to U . Consider the countable set including A,B, as well asall the sets Cγ = {δ ∈ Γ: γ · δ /∈ A} for γ ∈ B; we will show that it cannot bediagonalized. Suppose towards a contradiction that some condition p ∈ PΓ doesdiagonalize it. This means that there is m0 ∈ γ such that for all k > m ≥ m0,p(m) ∈ B and p(m) · p(k) ∈ A. There must be also some k0 ∈ ω such that forall k ≥ k0, p(k) ∈ Cp(m0). Then p(m0) · p(k0) should belong simultaneously toA and to the complement of A, which is a contradiction.


Finally, the statement of the classification theorem for the balanced virtualconditions in the poset PΓ is at hand.

Definition 7.4.6. Whenever 〈Γ, ·〉 is a countable semigroup and U is an ultra-filter on Γ, let τU denote the Coll(ω,P(Γ))-name for the (nonempty analytic)set of all conditions diagonalizing U .

Theorem 7.4.7. Let 〈Γ, ·〉 be a countable semigroup.

1. Whenever U is an idempotent ultrafilter then 〈Coll(ω,P(Γ)), τU 〉 is a bal-anced virtual condition;

2. every balanced pair is equivalent to 〈Coll(ω,P(Γ)), τU 〉 for some idempo-tent ultrafilter U ;

3. distinct idempotent ultrafilters give rise to inequivalent balanced virtualconditions.

In particular, the poset PΓ is balanced.

Proof. For (1), note that by Proposition 7.4.5, τU is a name for a nonempty setjsut in case U is an idempotent ultrafilter. (1) then follows immediately fromthe following claim:

Claim 7.4.8. Let U be an ultrafilter on Γ and 〈Q0, τ0〉, 〈Q1, τ1〉 are posets andtheir respective names for elements of PΓ diagonalizing U . Then Q0×Q1 τ0, τ1are compatible in PΓ.

Proof. Let q0 ∈ Q0 and q1 ∈ Q1 be conditions and n ∈ ω be a natural number.We must find an element γ ∈ Γ and conditions q′0 ≤ q0 and q′1 ≤ q1 and finitenonempty sets a0, a1 ⊂ ω with minimum larger than n such that q′0 γ =Πm∈a0τ0(m) and q′1 γ = Πm∈a1τ1(m). The compatibility of τ0, τ1 is thengranted by a genericity argument.

Let A0 ⊂ Γ be the set of all γ ∈ Γ such that there exists a finite nonemptyset a ⊂ ω with minimum greater than n and a condition q′ ≤ q0 in Q0 forcingγ = Πm∈aτ0(m). Since τ0 is forced to diagonalize U , it must be the case thatA0 ∈ U . Similarly, let A1 ⊂ Γ be the set of all γ ∈ Γ such that there exists afinite nonempty set a ⊂ ω with minimum greater than n and a condition q′ ≤ q1

in Q1 forcing γ = Πm∈aτ1(m). Since τ1 is forced to diagonalize U , it must bethe case that A1 ∈ U . Choose γ ∈ A0 ∩A1 and choose q′0 ≤ q0, q′1 ≤ q1 and setsa0, a1 witnessing the membership of γ in A0, A1. This completes the proof.

For (2), suppose that 〈Q, τ〉 is a balanced pair. Without loss of generalitywe may assume that Q collapses the size of P(Γ) ∩ V to ℵ0. Strengthening τin the Q-extension repeatedly by an application of Proposition 7.4.3, we mayassume that τ sorts out P(Γ)∩V . Let σ be the Q-name for the collection of allsets in P(Γ) ∩ V which τ accepts.

Claim 7.4.9. The membership of every set A ⊂ Γ in σ is decided by the largestcondition in Q.

7.4. SEMIGROUP ULTRAFILTERS 167

Proof. If not, then there is a set A ⊂ Γ and conditions q0, q1 ∈ Q such thatq0 A ∈ σ and q1 Γ \ A ∈ σ. Plainly, the condition 〈q0, q1〉 forces in theproduct Q×Q that τleft, τright are conditions incompatible in PΓ, contradictingthe balance assumption on the name τ .

Let U = {A ⊂ Γ: Q A ∈ σ}. This is an ultrafilter on Γ. By Propo-sition 7.4.5, it is an idempotent ultrafilter. By Claim 7.4.8, the pair 〈Q, τ〉 isequivalent to 〈Coll(ω,P(Γ)), τU 〉. (2) follows.

(3) is immediate. To prove the last sentence, note that for every conditionp ∈ PΓ, the Ellis–Numakura theorem yields an idempotent ultrafilter U suchthat for all n ∈ ω, the set {

∏m∈a p(m) : a ⊂ ω is a nonempty finite set and

min(a) > n} belongs to U . Then Coll(ω,P(Γ)) ΣτU ≤ p by the definition ofthe ordering PΓ and so 〈Coll(ω,P(Γ)), τU 〉 is a balanced virtual condition belowp.

Example 7.4.10. Let Γ be the group of finite subsets of ω with the symmetricdifference operation. Consider the poset PΓ below the natural initial conditionp ∈ Γω which assigns the singleton set {n} to each number n ∈ ω. The genericultrafilter added by the poset is known as the stable ordered union ultrafilter asintroduced in [10]. The stable union ultrafilters are studied in many places inthe literature, including the study of the PΓ-extension of the symmetric Solovaymodel [22].


Chapter 8

Other forcings

In this chapter we gather partial orders which do not fit in the previous chapters.

8.1 Chromatic numbers

Given a Polish space X, a Borel hypergraph Γ ⊂ [X]<ℵ0 , one can consider thetask of forcing a Γ-coloring c : X → ω. In principle, this is a very difficult taskand we do not have a general way of resolving it. The first poset one can consideris the poset PK where K is the Borel simplicial complex on X × ω consistingof finite partial Γ-colorings. The basic disadvantage of this poset is that notall maximal K-sets are total Γ-colorings, and there are elements of PK whichcannot be extended to total Γ-colorings. Thus, we have to restrict the posetsPK in some way to get the desired effect. This is not always straightforward,and the resulting posets are usually not simplicial complex posets in the senseof Definition 6.1.1. The most natural attempt is the following.

Definition 8.1.1. Let X be a Polish space and let Γ ⊂ [X]<ℵ0 be a Borel graphon a Polish space X. The Γ-coloring forcing PΓ is the partially ordered set ofall partial countable functions p : X → ω which can be extended to a Γ-coloringof the whole space X. The ordering is that of reverse inclusion.

It is immediate that the poset PΓ, if nonempty, forces the union of the condi-tions in the generic filter to be a total Γ-coloring with countably many colors.However, it is not at all clear whether the poset PΓ is Suslin, σ-closed, or bal-anced. In an interesting large class of Borel graphs Γ, these questions have asmooth resolution. Recall that a graph Γ on vertex set X has coloring numberκ if there is a well-ordering ≺ of X such that for each vertex x ∈ X the set{y ∈ X : y ≺ x and y Γ x} has cardinality smaller than κ [25]. This may seemlike a very complex notion especially in the case of infinite cardinals κ, but infact there is a simple characterization of analytic graphs of uncountable coloringnumber [1].

169

170 CHAPTER 8. OTHER FORCINGS

Theorem 8.1.2. Let Γ be a Borel graph on a Polish space X with countable col-oring number. Then PΓ contains a dense subset which is a Suslin ℵ0-distributiveposet. Moreover,

1. for every total Γ-coloring c : X → ω the pair 〈Coll(ω,X), c〉 is balanced;

2. for every balanced pair 〈Q, τ〉 there is a total coloring c such that 〈Q, τ〉 isequivalent to 〈Coll(ω,X), c〉;

3. distinct colorings yield inequivalent balanced conditions.

In particular, the poset PΓ is balanced.

Proof. Fix a well-ordering ≺ on the space X such that for every element x ∈ X,the set {y ∈ X : y ≺ x and {x, y} ∈ Γ} is finite. We start with a humbleobservation which will be used in several places below.

Claim 8.1.3. Let M be a countable elementary submodel of a large countablestructure containing ≺. Then every y ∈ X \M is Γ-connected with at mostfinitely many elements of X ∩M .

Proof. Suppose that y ∈ X \M is a point. Then for every x ∈ M ≺-greaterthan y, {x, y} /∈ Γ holds: if {x, y} ∈ Γ then y belongs to the finite set of allpoints ≺-smaller than x and connected with x and by elementarity it wouldhave to belong to the model M , which is not the case. Thus, the only points ofM connected to y in Γ are ≺-smaller than y, and there are only finitely manypoints like that by the definitory property of the ordering ≺.

Let P be the poset of all partial countable Γ-colorings p : X → ω such thatfor all points y ∈ X \ dom(p), the set {x ∈ dom(p) : {x, y} ∈ Γ} is finite. Theordering on P is that of reverse extension. The following two claims show thatP has the properties required in (1).

Claim 8.1.4. P is a Suslin ℵ0-distributive poset.

Proof. We first need to show that the set of all (enumerations of) conditions inP is Borel. Use Claim 8.1.3 to conclude that for every countable set a ⊂ X,the set b = {y ∈ X : ∃∞x ∈ a {x, y} ∈ Γ} is countable. By the Lusin–Novikovtheorem, there are Borel functions {fi : i ∈ ω} from Xω to X such that for everyz ∈ Xω, the list {fi(z) : i ∈ ω} includes all points in X which are Γ-connectedwith infinitely many points of rng(z). Now we can evaluate the complexity ofthe set of enumerations of conditions in the poset P . A function r : ω → X × ωis an enumeration of a condition in P just in case when rng(r) is a functionand a Γ-coloring, and, writing s : ω → X for the first component function of r,for every i ∈ ω if fi(s) is Γ-connected to infinitely many points in rng(s), thenfi(s) ∈ rng(s). This is a Borel condition.

To show that the poset P is Suslin, note that two conditions p0, p1 ∈ P arecompatible just in case they are compatible as functions: then p0 ∪ p1 will betheir lower bound. To argue for the ℵ0-distributivity, let p ∈ P be a condition

8.1. CHROMATIC NUMBERS 171

and let {Di : i ∈ ω} be a countable collection of open dense subsets of P . To finda condition q ≤ p in the intersection

⋂iDi, let M be a countable elementary

submodel of a large structure containing p,Γ,≺ and {Di : i ∈ ω} and use theelementarity of M to build a sequence pi for i ∈ ω by recursion so that

• p = p0 ≥ p1 ≥ p1 ≥ . . . are all conditions in M ;

• pi+1 ∈ Di and dom(pi+1) contains the i-th element of M ∩X in some fixedenumeration.

In the end, let q =⋃i pi and use Claim 8.1.3 to argue that q ∈ P as desired.

Claim 8.1.5. P is a dense subset of PΓ.

Proof. It is first necessary to show that P is a subset of PΓ. To this end, letp ∈ P . Let {bk : k ∈ ω} be a collection of pairwise disjoint infinite subsets ofω and c : X → ω be a total Γ-coloring. Let d : X → ω be the function definedas follows: d(x) = p(x) if x ∈ dom(p), and otherwise let d(x) be the smallestnumber in bc(x) which is different from the finitely many numbers {p(y) : y ∈dom(p) and {x, y} ∈ Γ}. It is not difficult to verify that d is a total Γ-coloringextending p, and so p ∈ Pω(Γ).

For the density, suppose that p ∈ PΓ is a condition. Let c : X → ω be atotal Γ-coloring extending p. Let M be a countable elementary submodel of alarge structure containing p, c, and ≺. By Claim 8.1.3, the set {y ∈ X : ∃∞x ∈M {x, y} ∈ Γ} is a subset of M . Thus, the function q = c �M is a condition inP extending p as desired.

For the classification of the balanced virtual conditions, we need a claim:

Claim 8.1.6. Let R0, R1 be posets and η0, η1 be respective R0- and R1-namesfor elements of X \ V . Then

1. R0 forces η0 to be Γ-connected with at most finitely many elements of theground model;

2. R0 ×R1 {η0, η1} /∈ Γ.

Proof. For the first item, suppose towards a contradiction that this fails asforced by some condition r ∈ R. Let M be a countable elementary submodel ofa large structure containing all relevant information, let g ⊂ R0 ∩M be a filtergeneric over the model M , and let y = η0/g ∈ X. By the forcing theorem, M [g]satisfies that y is Γ-connected with infinitely many elements of X ∩M . By theMostowski absoluteness for the model M [g], y is indeed connected with infinitelymany elements of M , and this contradicts the conclusion of Claim 8.1.3.

For the second item, suppose towards a contradiction that this fails as forcedby some condition 〈r0, r1〉 in the product. Let M ∈ N be countable elementarysubmodels of a large structure containing all relevant information. In the modelN , there are filters {gi : i ∈ ω} on the poset R0 ∩M which are mutually genericover the model M . By a mutual genericity argument applied in M , the points


xi = η0/gi ∈ X are pairwise distinct elements of N . Now, let h ⊂ R1 ∩M be afilter generic over the model N containing the condition r1 and let y = η1/h ∈X \N . By the forcing theorem in M , it is the case that M [Gi, h] |= {xi, y} ∈ Γfor each i ∈ ω. By the Mostowski absoluteness for the models M [gi, h], it isin fact the case that y is Γ-connected with each xi for i ∈ ω. This, however,contradicts Claim 8.1.3 applied to N .

Now, for item (1) of the theorem, let c : X → Γ be a total Γ-coloring. Notethat Coll(ω,X) c ∈ P by Claim 8.1.6(1). To see that the pair 〈Coll(ω,X), c〉is balanced, suppose that R0, R1 are posets and σ0, σ1 are R0- and R1-names forelements of P extending c; we must show that the product R0×R1 forces σ0, σ1

to be compatible in P ; in other words, σ0 ∪ σ1 is a function and a Γ-coloring.To verify that σ0∪σ1 is a function, use the product forcing theorem to concludethat dom(σ0)∩ dom(σ1) is forced to be a subset of the ground model, and bothσ0 � V and σ1 � V are forced to be equal to c. To see that σ0∪σ1 is a Γ-coloring,use Claim 8.1.3(2). This completes the proof of (1).

To argue for item (2), suppose that 〈Q, τ〉 is a balanced pair. Strengtheningτ if necessary we may assume that Q X ∩ V ∈ dom(p). By a balanceargument, for each x ∈ X there must be a specific number c(x) ∈ ω such thatQ τ(x) = c(x). It is not difficult to check that c : X → ω is a total Γ-coloringand Q× Coll(ω,X) τ ≤ c. (2) then follows from Proposition 5.2.4.

(3) is immediate. The last sentence now easily follows: if p ∈ PΓ is acondition then by the definition of PΓ it can be extended to a total Γ-coloringc : X → ω which then yields a balanced virtual condition below p by (1).

Example 8.1.7. Let n ≤ 3 be a number, let D ⊂ R be a countable set ofpositive reals converging to 0, and let ΓDn be the graph connecting two pointsof Rn if their distance belongs to D. The graph ΓDn has countable coloringnumber by [58, Theorem 7], and therefore the poset PΓDn is balanced.

Example 8.1.8. Let D ⊂ R be a countable set of positive reals and considerthe graph ΓD connecting two points of the plane if their distance belongs to D.The graph ΓD does not contain 2 times uncountable as a subgraph; by [25] itmust have countable coloring number and so the poset PΓD is balanced. If D isthe set of all positive rationals then the graph contains infinite cliques and soits chromatic number is infinite. If D is an algebraically independent set, thenit is not known whether the chromatic number is finite or infinite, and Bukh[14] conjectured the former.

Chromatic numbers of hypergraphs offer additional challenges. We provideelegant coloring posets for two natural classes of hypergraphs. They are lesscanonical than the rest of the posets in this book as they need a choice of anauxiliary tool: a Borel ideal J on ω containing all finite sets and such that Jis not generated by a countable subset of J . There are many such ideals, onepossible choice is the summable ideal.

Definition 8.1.9. Let X be a Borel abelian group. Let Φ be a countable fieldwhose multiplicative group acts on X in a Borel way, making X into a vector


space. The 3-Hamel hypergraph is the set Γ ⊂ [X]3 consisting of triples oflinearly dependent nonzero elements of X. A Γ-coloring c : X → ω is called an3-Hamel decomposition.

Proposition 8.1.10. Let X be a Borel vector space over a countable field Φ.Then there is a 3-Hamel decomposition of X.

Proof. Let A ⊂ X be a Hamel basis. Let B a countable basis of the topology ofthe ambient Polish space of which X is a Borel subset. We define a Γ-coloringwhose range is the countable set C = (Φ × B)<ω. For each nonzero elementx ∈ X, let x = Σi∈nφiyi be the unique expression of x as a linear combinationof basis vectors with nonzero coefficients, in which the basis vectors are listedin an increasing order with respect to some fixed Borel linear ordering of X.Let 〈Oi : i ∈ n〉 be the first sequence (in some fixed enumeration) of pairwisedisjoint basic open sets in the basis B such that yi ∈ Oi holds for every i ∈ n.Define the coloring c : X → C by setting c(0) = 0 and c(x) = 〈φi, Oi : i ∈ n〉where φi, Oi are as above.

We must show that there is no monochromatic Γ-edge. Suppose towards acontradiction that x0, x1, x2 ∈ X are distinct nonzero elements of X, all withthe same color 〈φi, Oi : i ∈ n〉, such that some nontrivial linear combination ofthem yields zero. Let y0

i ∈ Oi be the basis vectors yielding x0, similarly for y1i

and y2i . Since the three vectors are distinct, there must be i ∈ n and j ∈ 3 such

that yji is distinct from yki for all k ∈ 3 different from j. But then, the basis

vector yji can be expressed as a linear combination of all the remaining basisvectors used to obtain the points x0, x1, x2. Since the remaining vectors are alldistinct from yji , this is a contradiction.

We now provide a balanced forcing which adds a 3-Hamel coloring to a givenBorel vector space over a countable field.

Definition 8.1.11. Let X be a Borel vector space over a countable field Φ. The3-Hamel decomposition forcing P is the set of all functions p such that dom(p)is a countable subspace of X, rng(p) ⊂ ω, and p is a Γ-coloring on dom(p) whereΓ is the 3-Hamel hypergraph. The ordering is defined by q ≤ p if p ⊂ q, and forevery dom(p)-coset e ⊂ dom(q) distinct from dom(p), the image q′′

⋃φ∈Φ φ · e

belongs to the ideal J .

It is not difficult to see that P is a σ-closed Suslin poset. We now classify itsbalanced conditions.

Theorem 8.1.12. Let X be a Borel vector space over a countable field Φ. LetΓ be the 3-Hamel hypergraph and let P be the 3-Hamel decomposition forcing.

1. If c : X → ω is a Γ-coloring, the pair 〈Coll(ω,X), c〉 is balanced;

2. if 〈Q, τ〉 is a balanced pair, then there is a Γ-coloring c : X → ω such thatthe balanced pairs 〈Q, τ〉 and 〈Coll(ω,X), c〉 are equivalent;

3. distinct Γ-colorings yield inequivalent balanced pairs.


In particular, P is balanced.

Proof. For (1), let c : X → ω be a Γ-coloring. Let V [G0], V [G1] be mutuallygeneric extensions in which the set X ∩ V is countable and let p0 ∈ V [G0],p1 ∈ V [G1] be conditions below c; we must produce a lower bound q ≤ p0, p1

in the poset P in the model V [G0, G1]. Write d0 = dom(p0 and d1 = dom(p1).For every point x ∈ X \ (d0 ∪ d1), write e0(x) = {y0 ∈ d0 : ∃y1 ∈ d1 x is a linearcombination of y0, y1} and e1(x) = {y1 ∈ d1 : ∃y0 ∈ d0 x is a linear combinationof y0, y1}. The following claim is key.

Claim 8.1.13. Let x ∈ X \ (d0∪d1) be any point. If the set e0(x) is nonempty,then there is a V -coset f distinct from V such that e0 =

⋃φ∈Φ φ·f , and similarly

for e1(x).

Proof. Fix the point x and write e0(x) = e0 and e1(x) = e1. We argue for e0,the case of e1 is symmetric. First note that if y0 ∈ e0 then y0 cannot belong toV , as in such a case x would be a linear combination of elements of d1 and soan element of d1, contradicting the initial choice of x.

It is clear that the set e0 is closed under scalar multiplication and additionof vectors in V . To show that e0 is obtained from a single V -coset, suppose thaty0, z0 ∈ e0 are distinct points; we must show that there is a linear combinationof y0, z0 which yields a point in V . Then there must be points y1, z1 ∈ e1 andfield elements φ0, phi1, ψ0, ψ1 such that x = φ0y0 + φ1y1 and x = ψ0z0 + ψ1z1.It follows that φ0y0 − ψ0z0 = ψ1z1 − φ1y1. The left-hand side of the equationbelongs to V [G0], the right hand side belongs to V [G1], and by the productforcing theorem, their common value v must belong to V . But then v = φ0y0−ψ0z0 is the desired linear combination.

Now, let d be any countable subspace of X containing both d0, d1 as subsets.Write a = d \ (d0 ∪ d1). Let I be the set {p′′0e0(x) : x ∈ a} united with{p′′1e1(x) : x ∈ a}. By the claim and the definition of the ordering on P , Iis a countable set of sets in the ideal J . By the initial assumptions on the idealJ , there is a set b ∈ J which cannot be covered by finitely many elements of Iand a finite set. The set a can be easily decomposed into infinitely many setswith the same property, and so there are pairwise disjoint sets b(x) ⊂ ω forx ∈ a such that none of them can be covered by finitely many elements of I anda finite set, and such that

⋃x b(x) ∈ J . Let q : d→ ω be any function such that

p0, p1 ⊂ q, and for every point x ∈ a we have q(x) ∈ b(x) \ (p′′0e0(x) ∪ p′′1e1(x)).We claim that q ≤ p0, p1 is the desired lower bound.

To see that q ∈ P , we must show that q is a Γ-coloring. Suppose towards acontradiction that {x0, x1, x2} ⊂ d is a monochromatic triple of nonzero linearlydependent pairwise distinct points. The triple cannot be a subset of d(p0) ord(p1) since p0, p1 are Γ-colorings. The triple cannot use more than one pointin a since q � a is an injection. We are left with only one case, where (afterre-indexing if necessary) x2 ∈ a and x0 ∈ d0, x1 ∈ d1. In such a case, x0 ∈ e0(x)and x1 ∈ d1(x) holds and by the initial choice of the function q, q(x2) is distinctfrom both q(x0) and q(x1).


To see that q ≤ p0, let e be a d0-coset distinct from d0; we must show thatthe set q′′

⋃φ∈Φ φ · e belongs to J . By the product forcing theorem, e ∩ d1 is in

fact a V -coset distinct from V , and so q′′⋃φ∈Φ φ ·e∩d1 = p′′1

⋃φ∈Φ φ ·e∩d1 ∈ J

since p1 ≤ c. Thus, the set q′′⋃φ∈Φ φ · e is covered by the union of the J-small

sets q′′⋃φ∈Φ φ · e ∩ d1 and b and therefore belongs to J . The case q ≤ p1 is

symmetric.For (2), suppose that 〈Q, τ〉 is a balanced pair. Strengthening τ if necessary,

we may assume that Q (X ∩ V ) ⊂ dom(τ). By a balance argument, for eachpoint x ∈ X ∩ V there is a number c(x) ∈ ω such that Q τ(x) = c(x). It isimmediately clear that the map c is a Γ-coloring. We will show that the balancedpairs 〈Q, τ〉 and 〈Coll(ω,X), c〉 are equivalent. To this end, it is enough to arguethat Q τ ≤ c. Suppose towards a contradiction that some condition q ∈ Qforces the opposite. Let G0, G1 ⊂ Q be mutually generic filters meeting thecondition q, and write p0 = τ/G0 and p1 = τ/G1. To reach the contradictionwith the balance of the pair 〈Q, τ〉, argue that p0, p1 are conditions incompatiblein P . To see this, use the forcing theorem and the assumption that p0 6≤ c tofind a V -coset e ⊂ dom(p0) distinct from V such that p′′0

⋃φ∈Φ φ · e /∈ J . Now,

the set e is a subset of a single dom(p1)-coset distinct from dom(p1) by theproduct forcing theorem. Therefore, it prevents finding a common lower boundof p0, p1.

(3) is clear. For the last sentence, we prove a claim which will come handylater.

Claim 8.1.14. Let p ∈ P be a condition and a ⊂ ω be an infinite set. Thenthere is a Γ-coloring c : X → ω such that p ⊂ c and all points not in dom(p) geta c-color from the set a.

Proof. Use Proposition 8.1.10 to find a coloring d : [X]2 → ω. Fix a collection{an : n ∈ ω} of pairwise disjoint infinite sets such that

⋃n an ∈ a. Consider the

graph ∆ on X \ dom(p) connecting points x, y if there is a point z ∈ dom(p)such that the set {x, y, z} is linearly dependent. Note that ∆ is a locally count-able graph, and so its path-connectedness equivalence relation E has all classescountable. Therefore, one can find a function c : X → ω such that p ⊂ c, forall x ∈ X \ dom(p) c(x) ∈ ad(x) holds, and c restricted to any E-class is aninjection. To check that c is a Γ-coloring, assume that {x0, x1, x2} ∈ Γ is atriple of pairwise distinct points. If all three of them belong to dom(p) then thethe triple is not monochromatic as p is a Γ-coloring. If two of them belong todom(p) then so does the third one by the closure properties of dom(p) and weare in the previous case. If exactly one of the points belongs to dom(p) thenthe other two are E-related and assigned a different color by c. Finally, if noneof the points are in dom(p) then the triple is not monochromatic since d is aΓ-coloring.

Now, let p ∈ P be a condition. Choose an infinite set a ∈ J and find a Γ-coloringc : X → ω such that p ⊂ c and all points not in dom(p) get a c-color from theset a. Clearly, Coll(ω,X) c ≤ p and so c is the desired balanced conditionstronger than p.


Example 8.1.15. Ceder [17] provided a countable subfield Φ of the complexplane C such that any 3-Hamel decomposition of the complex plane over Φconsists of pieces which contain no equilateral triangles.

A very similar poset can produce a decomposition of a vector space into count-ably many internally linearly independent sets.

Definition 8.1.16. Let X be a Borel vector space over a countable field Φ.The Hamel hypergraph is the set Γ ⊂ [X]<ℵ0 consisting of linearly dependentfinite sets of nonzero elements of X. A Γ-coloring c : X → ω is called an Hameldecomposition.

Proposition 8.1.17. Let X be an uncountable Borel vector space over a count-able field Φ. Then there is a Hamel decomposition of X if and only if theContinuum Hypothesis holds.

Proof. Assume first that the Continuum Hypothesis holds. Exhaust X by aninclusion increasing sequence of countable subspaces, X =

⋃α∈ω1

Xα. Letc : X → ω be a map which is an injection on each set Xα \

⋃β∈αXβ . We

claim that c is a Hamel decomposition. Indeed, suppose that a ⊂ X is a finitelinearly dependent set. Fix a linear combination of elements of a adding up to 0.Let α ∈ ω1 be the largest ordinal such that the set Xα \

⋃β∈αXβ contains some

element of a with a nonzero coefficient in the linear combination. This elementcannot be unique, since then it would be expressible as a linear combination ofthe subspace

⋃β∈αXβ , an impossibility. So a contains at least two elements in

the set Xα \⋃β∈αXβ ; thus, a cannot be monochromatic, as c is injective on

this set.On the other hand, suppose that the Continuum Hypothesis does not hold,

and let c : X → ω is a map; we must produce a nontrivial monochromaticlinear combination adding up to zero. Let M be an elementary submodel ofcardinality ℵ1 of a large structure, containing the map c. Use the failure ofthe Continuum Hypothesis to produce a point x0 ∈ X \M . Use a countingargument to find distinct points y0, y1 ∈ X ∩M and a number n ∈ ω such thatc(x0 − y0) = c(x0 − y1) = n. Use the elementarity of the model M to finda point x1 ∈ X ∩M such that c(x1 − y0) = c(x1 − y1) = n. Note that thepoints z0 = x0 − y0, z1 = x0 − y1 and z2 = x1 − y0, z3 = x1 − y1 are pairwisedistinct, they attain the same color n, and z0 − z1 − z2 + z3 = 0. The proof iscomplete.

Definition 8.1.18. Let X be a Borel vector space over a countable field Φ. TheHamel decomposition forcing P is the set of all functions p such that dom(p) is acountable subspace of X, rng(p) ⊂ ω, and p is a Γ-coloring on dom(p) where Γ isthe Hamel hypergraph. The ordering is defined by q ≤ p if p ⊂ q, and there is nonontrivial linear combination of a q-monochromatic set elements of dom(q) withresult in dom(p), and for every dom(p)-coset e ⊂ dom(q) distinct from dom(p),the set {n ∈ ω : there exists a linear combination of a q-monochromatic set withresult in

⋃φ∈Φ φ · e} belongs to the ideal J .

8.2. DISCONTINUOUS HOMOMORPHISMS 177

It is not difficult to see that P is a σ-closed Suslin poset. We now classify itsbalanced conditions.

Theorem 8.1.19. Let X be a Borel vector space over a countable field Φ. LetΓ be the Hamel hypergraph and let P be the Hamel decomposition forcing.

1. If c : X → ω is a Γ-coloring, the pair 〈Coll(ω,X), c〉 is balanced;

2. if 〈Q, τ〉 is a balanced pair, then there is a Γ-coloring c : X → ω such thatthe balanced pairs 〈Q, τ〉 and 〈Coll(ω,X), c〉 are equivalent;

3. distinct Γ-colorings yield inequivalent balanced pairs.

In particular, the Continuum Hypothesis is equivalent to the statement that Pis balanced.

As the proof of Theorem 8.1.19 is nearly literally the same as the one for The-orem 8.1.12, we dare to leave it to the patient reader.

Example 8.1.20. Let X be the (non-Polishable) group of finite subsets of anuncountable Polish space X with the symmetric difference operation. View Xas a vector space over the binary field. Let c : X → ω be a Hamel decompositionof X. The restriction c � [Y ]2 → ω is a decomposition of the clique graph on Yinto countably many sets with no monochromatic cycle.

8.2 Discontinuous homomorphisms

Another task which can be handled by balanced posets derived from simplicialcomplexes is the adding of discontinuous homomorphisms between topologicalstructures. In this section, we treat the case of Polish groups. The problemof the existence of discontinuous homomorphisms from one Polish group to an-other has been studied for decades. In ZFC, some groups such as the unitarygroup [95] or the group of homeomorphisms of the unit circle [77] cannot behomomorphically mapped to another Polish group in a discontinuous way. Ina ZF+DC context, existence of discontinuous homomorphisms has interestingconsequences: in general, it implies that the chromatic number of the Hamminggraph H2 is 2 [76], for some specific groups it implies the existence of a nonprin-cipal ultrafilter [90, Theorem 4.1]. For an exposition, see [75]. In this section,we provide a balanced forcing adding a discontinuous homomorphism of Polishgroups in a certain common special case.

Definition 8.2.1. Let Γ and ∆ be Polish groups. The simplicial complexK(Γ,∆) on X × Y consists of all finite sets a ⊂ X × Y which are subsetsof a homomorphism from X to Y . We will write P (Γ,∆) for the associatedΓ,∆-homomorphism poset of countable K(Γ,∆)-sets ordered by inclusion.

As it was the case in Section 8.1, it seems to be difficult to evaluate the com-plexity of the complex K(Γ,∆) without having additional information on thenature of the two groups. In this section, we will deal with a very special casewhich nevertheless resolves some interesting problems.


Theorem 8.2.2. Let Γ,∆ be abelian Polish groups on respective Polish spacesX,Y . Suppose that ∆ is divisible. Then the poset P (Γ,∆) is Suslin and

1. for every homomorphism h : Γ → ∆, the pair 〈Coll(ω,X × Y ), h〉 is bal-anced;

2. for every balanced pair 〈Q, τ〉 there is a homomorphism h : Γ → ∆ suchthat the pairs 〈Q, τ〉 and 〈Coll(ω,X × Y ), h〉 are equivalent;

3. distinct homomorphisms yield inequivalent balanced virtual conditions.

In particular, the poset P (Γ,∆) is balanced.

Proof. Write K = K(Γ,∆) and P = P (Γ,∆). For the Suslinness of P , it isenough to argue that the simplicial complex K is Borel. For that, note thata finite partial function a ⊂ X × Y belongs to K if and only if for every Z-combinations Σiniγi = Σjnjγj of elements of P , it is the case that Σinia(γi) =Σjnja(γj). The left-to-right implication is clear, since the right hand side holdsfor every homomorphism from Γ to ∆. For the right-to-left implication, notethat the assumption on the right hand side implies that a can be extended to ahomomorphism from the subgroup generated by dom(a) to ∆, which then canbe extended to a homomorphism of all of Γ to ∆ by the divisibility assumptionon ∆ and Baer’s criterion [3].

For (1), suppose that V [H0], V [H1] are mutually generic extensions of theground model V and a0, a1 ∈ K are finite functions in the respective models suchthat a0 ∪ h and a1 ∪ h are K-sets. We need to show that a0 ∪ a1 ∈ K. Towardsa contradiction, suppose that this fails, and write a = a0 ∪ a1. By the workof the previous paragraph, there must be a Z-combinations Σiniγi = Σjnjγjsuch that Σinia(γi) 6= Σjnja(γj). Note that elements of both a0, a1 must beused in these combinations. Rearranging the combinations, we may assumethat the left combination uses only elements from a0 and the right one usesonly elements of a1. By the product forcing theorem, γ = Σiniγi = Σjnjγjmust belong to V . Since both a0 and a1 form a K-set with the homomorphismh, the outputs Σinia(γi),Σjnja(γj) must both be equal to h(γ), and thereforecannot be distinct, a contradiction.

This also means that P forces that the union of the generic filter is a totalhomomorphism from Γ to ∆; the name for the union we will denote by hgen. For(2), let 〈Q, τ〉 be a balanced pair. Strengthening τ if necessary, we may assumethat for each point γ ∈ Γ ∩ V Q τ(γ) ∈ dom(τ). Now, fix a point γ ∈ Γ.By a balance argument, for every basic open set O ⊂ ∆ it must be the casethat either Q τ(γ) ∈ O or Q τ(γ) /∈ O. This means that there is a singlepoint δ(γ) ∈ ∆ which is in the intersection of all open sets O ⊂ ∆ for which theformer option prevails; it must be the case that Q τ(γ) = δ(γ). The functionh : γ 7→ δ(γ) must be a homomorphism from Γ to ∆. Clearly Q τ ≤ h; byProposition 5.2.4, the pairs 〈Q, τ〉 and 〈Coll(ω,X × Y ), h〉 are equivalent.

Finally, (3) is obvious. For the last sentence, note that every conditionp ∈ P can be extended to a homomorphism from Γ to ∆ by the work of the firstparagraph.

8.3. AUTOMORPHISMS OF P(ω) MODULO FINITE 179

Example 8.2.3. Suppose that Γ,∆ are abelian Polish groups, with Γ tor-sion free and uncountable and ∆ divisible–the case Γ = R and ∆ = R/Z isof particular interest. Then the poset P = P (Γ,∆) forces the generic homo-morphism to be discontinuous. To see this, let p ∈ P be a condition. Let〈γi : i ≤ ω〉 be a sequence of elements of Γ which are linearly independentover dom(p) and such that lim γi = γω. Pick distinct elements δ, δω and letq = p ∪ {〈γi, δ〉 : i ∈ ω, 〈γω, δω〉}. It is immediate that q is a K(Γ,∆)-set, and itforces γω to be a point of discontinuity of the generic homomorphism.

8.3 Automorphisms of P(ω) modulo finite

Automorphisms of the Boolean algebra P(ω) modulo finite have been investi-gated for decades. The simply definable ones are trivial [98] and the ProperForcing Axiom implies that downright all automorphisms of the algebra aretrivial [83]. It turns out that there is a natural balanced poset for adding anontrivial automorphism of the Boolean algebra B = P(ω) modulo finite.

Definition 8.3.1. The automorphism poset P consists of pairs p = 〈Bp, πp〉where Bp ⊂ B is a countable subalgebra and πp is its automorphism. Theordering is that of coordinatewise reverse inclusion.

The automorphism poset is clearly σ-closed and Suslin. By a simple densityargument, it adds a nontrivial automorphism of the algebra B. This is animmediate consequence of the following observation.

Fact 8.3.2. [9, Theorem 2.3] Let C ⊂ B be a countable algebra and π : C → Cbe an automorphism. Let χ : B → B be an automorphism. Then there is anautomorphism η : B → B extending π and not equal to χ.

The balanced virtual conditions in the poset P are naturally classified by au-tomorphisms of the algebra B. Let π : B → B be such an automorphism, andconsider the Coll(ω,B)-name τπ for the (set of all conditions stronger than the)condition 〈BV , π〉 ∈ P . The following theorem provides the key classificationinformation.

Theorem 8.3.3. Let P be the automorphism poset.

1. For every automorphism π of B, the pair 〈Coll(ω,B), τπ〉 is balanced inP ;

2. for every balanced pair 〈Q, τ〉 there is an automorphism π of B such thatthe balanced pairs 〈Q, τ〉 and 〈Coll(ω,B), τπ〉 are equivalent;

3. distinct automorphisms yield inequivalent balanced conditions.



Proof. Item (1) is actually the most demanding part. Suppose that V [H0],V [H1] are mutually generic extensions of V , and p0 = 〈B0, π0〉 ∈ V [H0] andp1 = 〈B1, π1〉 ∈ V [H1] are Boolean algebras extending BV and automorphismsextending π respectively. Let C be the subalgebra of BV [H0,H1] generated byB0 ∪ B1. We will find an automorphism of C extending π0 ∪ π1, providing acommon lower bound of the conditions p0, p1.

Claim 8.3.4.

1. If a0 ∈ B0, a1 ∈ B1 and c ∈ BV are such that a0 ∧ a1 ≤ c, then π0(a0) ∧π1(a1) ≤ π(c) holds;

2. If a0, b0 ∈ B0 and a1, b1 ∈ B1 and a0∧a1 ≤ b0∧b1, then π0(a0)∧π1(a1) ≤p0(b0) ∧ π1(b1) holds;

3. If a0, bi0 for i ∈ n belong to B0, a1, b

i0 for i ∈ n belong to B1, and a0∧a1 ≤∨

i(bi0 ∧ bi1) holds, then π0(a0) ∧ π1(a1) ≤

∨(π0(bi0) ∧ π1(bi1)) holds.

Proof. For (1), note that a0 − c and a1 − c are disjoint sets in the mutuallygeneric models V [H0] and V [H1], and so by Proposition 1.7.8 there are groundmodel sets d0, d1 such that a0 − c ≤ d0, a1 − c ≤ d1, and d0 ∧ d1 = 0. Thenπ0(a0)− π(c) ≤ π(d0) since π0 is an automorphism of B0 extending π. For thesame reason π1(a1) − π(c) ≤ π(d1) holds; moreover, π(d0) ∧ π(d1) = 0 followsfrom the fact that π is an automorphism. For (2), the inequality a0∧a1 ≤ b0∧b1yields (a0 ∧ b0) ∧ (a1 − b1) = 0. Since a0 ∧ b0 ∈ V [H0] and a1 − b1 ∈ V [H1]and the two extensions are mutually generic, by Proposition 1.7.8 there has tobe a ground model set c ⊂ ω such that a0 ∧ b0 ≤ c and c ∧ (a1 − b1) = 0.For the same reason, there is a ground model set d ⊂ ω such that a1 ∧ b1 ≤ dand d ∧ (a0 \ b0) = 0. The choice of c shows that a0 ∧ a1 ≤ c and moreovera1∧c ≤ b1; for the same reason, a0∧a1 ≤ d and a0∧d ≤ b0 holds. In total, writinge = c ∧ d, we have a0 ∧ a1 ≤ e, a0 ∧ e ≤ b0, and a1 ∧ e ≤ b1. By the first item,π0(a0) ∧ π1(a1) ≤ π(e) follows. Moreover, since π0 and π1 are automorphisms,the inequalities π0(a0) ∧ π(e) ≤ π0(b0) and π1(a1) ∧ π(e) ≤ π1(b1) hold. Theconclusion of (2) is then at hand.

For (3), for each nonempty set t ⊂ n let ct =∧i∈t b

i0 −

∨i/∈t b

i0; these are

pairwise disjoint sets in V [H0]. Write also dt =∨i∈t b

i1; these are sets in V [H1].

Now, a0 ∧ a1 ⊂∨t ct, and by (2) π0(a0) ∧ π1(a1) ≤

∨t π0(ct) holds. For each

t ⊂ n, a0∧a1∧ ct ⊂ dt holds, and by (2) π0(a0)∧π1(a1)∧π0(ct) ⊂ π1(dt) holds.The conclusion of (3) follows.

Now, express each element of the algebra C as a disjunction of conjunctions ofelements of B0 and B1, and define χ(

∨i(a

i0 ∧ ai1)) =

∨i(π0(ai0) ∧ π1(ai1)) where

ai0 ∈ B0 and ai1 ∈ B1. Claim 8.3.4(3) shows that the definition of χ(c) does notdepend on the choice of the representation of the element c ∈ C as a disjunctionof conjunctions. The map χ : C → C preserves ordering by (3) of Claim 8.3.4and therefore is an automorphism of the algebra C as desired.

For item (2) of the theorem, extend τ if necessary so that Q τ = 〈C,χ〉 forsome algebra C containing BV and some automorphism χ of C. By a balance

8.4. KUREPA FAMILIES 181

argument, for each element b ∈ B there must be an element π(b) ∈ B suchthat Q χ(b) = π(b). It is immediate that π is an automorphism of B andQ 〈C,χ〉 ≤ 〈BV , π〉. Item (2) then follows from (1) and Proposition 5.2.4.

Finally, item (3) is obvious. For the last sentence, apply (1) with Fact 8.3.2.

8.4 Kurepa families

The notion of a Kurepa family on a set is an old one [61]; it appears intermit-tently in modern set theory [94], [24, Section 7]. In this section, we show howto force a cofinal Kurepa family on a Polish space with a balanced poset.

Definition 8.4.1. Let X be a set.

1. A Kurepa family is a set A ⊂ [X]ℵ0 such that for every countable setb ⊂ X, the set {a ∩ b : a ∈ A} is countable;

2. the family is cofinal if for every countable set b ⊂ X there is a set a ∈ Asuch that b ⊂ a.

The main task of this section is to show how a cofinal Kurepa family for a givenPolish space can be added by balanced forcing.

Definition 8.4.2. Let X be an uncountable Polish space. The Kurepa poset Pis a poset of all countable sets p ⊂ [X]ℵ0 closed under finite intersections. Theordering is defined by q ≤ p if p ⊂ q and for every a ∈ q and b ∈ p, a ∩ b ∈ p.

Clearly the poset P does not depend on the choice of the uncountable Polishspace X up to isomorphism. Clearly, P is a σ-closed Suslin forcing. If G ⊂ P isa generic filter, then a simple genericity argument shows that

⋃G is a cofinal

Kurepa family on X which is closed under finite intersections. For every setA ⊂ P(X) closed under finite intersections note that A is a Coll(ω,P(X))-name for an element of the Kurepa poset. It turns out that if X ∈ A then thepair 〈Coll(ω,P(ω)), A)〉 is balanced, and all balanced pairs are up to equivalenceof this form. This is the content of the following theorem.

Theorem 8.4.3. Let X be an uncountable Polish space and let P be the Kurepaposet on X.

1. if A ⊂ P(X) is a set containing X and closed under finite intersections,then the pair 〈Coll(ω,P(X)), A〉 is balanced;

2. if 〈Q, τ〉 is a balanced pair for P then there is a set A ⊂ P(X) closedunder finite intersections and containing X such that the balanced pairs〈Q, τ〉 and 〈Coll(ω,P(X)), A〉 are equivalent;

3. distinct sets as in (1) yield inequivalent balanced conditions.



Proof. For (1), fix a set A ⊂ P(X) closed under finite intersections and con-taining the set X. Suppose that V [H0], V [H1] are mutually generic extensionsof the ground model V in which the set P(X) ∩ V is countable. Suppose thatp0 ∈ V [H0] and p1 ∈ V [H1] are conditions stronger than A; we must showthat p0, p1 ∈ P are compatible, in other words that p0 ∪ p1 ∈ P . To thisend, let a0 ∈ p0 and a1 ∈ p1 be sets; we must show that a0 ∩ a1 ∈ p0 ∩ p1.To see this, first the product forcing theorem shows that a0 ∩ a1 ⊂ V ∩ Xholds. Second, both sets a0 ∩ V ∩X and a1 ∩ V ∩X must belong to A, sinceV ∩X ∈ A and A ≤ p0, p1. Since A is closed under finite intersections, we seethat a0∩a1 = (a0∩V )∩ (a1∩V ) ∈ A ⊂ p0∩p1 holds. Therefore, the conditionsp0, p1 are compatible as desired.

(2) is more challenging. Let 〈Q, τ〉 be a balanced pair in the poset P . With-out loss of generality we may assume that Q |P(X)∩V | = ℵ0. Strengtheningthe condition τ repeatedly in the Q-extension if necessary, we may assume thatfor each set a ⊂ X in the ground model, either a ∈ τ or there is b ∈ τ suchthat a ∩ b /∈ τ . A balance argument shows that for each set a ⊂ X in theground model, the largest condition in Q decides which alternative prevails.Let A ⊂ P(X) be a set of all sets a ⊂ X for which the former alternative pre-vails. We will show that A is a set closed under finite intersections containingX, and Q A ≥ τ . The desired equivalence of 〈Q, τ〉 and 〈Coll(ω,P(X)), A〉then follows from Proposition 5.2.4.

The closure of the set A under finite intersections follows immediately fromthe fact that Q forces τ to be closed under finite intersections. To see thatX ∈ A holds, suppose towards a contradiction that it fails; so there must be aQ-name σ for a subset of X such that Q σ ∈ τ and σ∩V /∈ τ . Strengthen thename τ if necessary to contain a set which contains X ∩ V as a subset, and letη be a Q-name for such a set in τ . Let H0, H1 ⊂ Q be mutually generic filters,let p0 = τ/H0 and p1 = τ/H1, and let a0 = η/H0 and a1 = σ/H1. By a mutualgenericity argument, a0 ∩ a1 = (a0 ∩ V ) ∩ (a1 ∩ V ) = a1 ∩ V . It follows thata0 ∩ a1 /∈ p1 and so p0, p1 are incompatible conditions in P , contradicting thebalance assumption.

Finally, to show that Q τ ≤ A, suppose towards a contradiction that thisfails. Passing to a condition in Q if necessary we may find a set b ∈ A and aQ-name σ such that Q σ ∈ τ and σ ∩ b /∈ A. Now proceed similarly to theprevious paragraph. Strengthen the name τ if necessary to contain a set whichcontains X ∩ V as a subset, and let η be a Q-name for such a set in τ . LetH0, H1 ⊂ Q be mutually generic filters, let p0 = τ/H0 and p1 = τ/H1, and leta0 = η/H0 and a1 = σ/H1. By a mutual genericity argument, a0 ∩ b ∩ a1 =(a0 ∩ b) ∩ (a1 ∩ b) = a1 ∩ b. Now, if a1 ∩ b ∈ V , then a1 ∩ b ∈ A, contradictingthe choice of the name σ. Thus, it must be the case that a1 ∩ b /∈ V , and by amutual genericity argument a1∩b /∈ V [H0]. It follows that a0∩b∩a1 /∈ p0 and sop0, p1 are incompatible conditions in P , contradicting the balance assumption.

Finally, (3) is obvious. The balance of the poset P follows from (1): if p ∈ Pis a condition then 〈Coll(ω,X), p ∪ {X ∩ V }〉 is a balanced pair below p.

8.5. SET MAPPINGS 183

8.5 Set mappings

In this section, we develop posets for adding interesting set mappings. Letn ∈ ω. A set mapping (of arity n) on a set X is just a function f : [X]n → P(X)such that f(a) ∩ a = 0 holds for every set a ∈ [X]n. A free set for f is a setb ⊂ X such that f(a) ∩ b = 0 holds for all a ∈ [b]n. Set mappings withoutlarge free sets have been investigated by Komjath ??? and others. It is possibleto add such mappings on Polish spaces by balanced forcing. We start with aninteresting limiting result.

Proposition 8.5.1. (ZF) Let X be an uncountable Polish space. If there is a setmapping f : [X]2 → X<ℵ0 without a free triple, then there is a countable-to-onefunction from X to ω1.

Proof. For each x ∈ X let Mx be the model of sets hereditarily ordinally defin-able from x and f . Let ≤ be the pre-ordering on the space X defined by y ≤ xif y ∈Mx.

First, we claim that ≤ is linear. To prove it, suppose towards a contradictionthat x0, x1 ∈ X are ≤-incomparable elements. Let x2 ∈ X be an ordinallydefinable point which does not belong to the finite set f(x0, x1). Then x1 /∈f(x0, x2) since x1 is not definable from x0, and x0 /∈ f(x1, x2) since x0 is notdefinable from x1. Thus, the set {x0, x1, x2} is a free triple for f , contradictingthe initial choice of f .

Second, we claim that if y < x are points in X then Mx |= My ∩ X isa countable set. Suppose towards a contradiction that this fails. Working inthe model Mx, let N be a countable elementary submodel of a large structurecontaining x, y and f � Mx. Since My contains uncountably many reals fromthe point of view of Mx, there is a point x0 ∈My \N . Let x1 ∈My ∩N be anypoint which does not belong to f(x, x0). As in the first paragraph, {x, x0, x1}is a free triple for f , contradicting the initial choice of f .Case 1. There is a point x such that Mx contains uncountably many elementsof X. By the previous two paragraphs, Mx contains all points of X. As ZFCproves that existence of a set mapping with finite values and no free triple isequivalent to the continuum hypothesis and Mx is a model of ZFC containing fas an element, we conclude that Mx contains an injection from X to ωMx

1 = ω1.This proves the proposition in this case.Case 2. Case 1 fails. In this case, let h be the map on X defined by h(x) = ωMx

1 .By the failure of Case 1, the range of h is a subset of ω1. It is also a countable-to-one map. Suppose towards a contradiction that for some ordinal α ∈ ω1, theset A = {y ∈ X : h(y) = α} is uncountable. Let y ∈ A be any point, and usethe case assumption to find y ∈ A such that x /∈ My. By the linearity of ≺, itfollows that y < x. By the previous work, the reals of My are a countable setin Mx, violating the assumption that the two models have the same ω1. Theproposition follows in this case as well.

Proposition 8.5.1 shows that we cannot hope to produce a set mapping withfinite values with no free triple, as such a mapping yields an uncountable se-


quence of F2-classes. However, other options for set mappings are achievable inbalanced extensions.

Definition 8.5.2. Let X be an uncountable Polish space. The fat set mappingforcing is the poset P of all functions p such that for some countable set d(p) ⊂X, p : [d(p)]2 → P(d(p)) is a set mapping without free triples. The ordering isdefined by q ≤ p if d(p) ⊂ d(q) and p ⊂ q.

It is clear that P is a σ-closed Suslin forcing. The union of the P -generic set ifa set mapping with countable values and no free triples.

Theorem 8.5.3. Let P be the fat set mapping forcing.

1. If f : [X]2 → P(X) is any set mapping without free triples, then 〈Coll(ω,X), f〉is a balanced pair;

2. if 〈Q, τ〉 is any balanced pair, there is a set mapping f : [X]2 → P(X)without free triples such that the balanced pairs 〈Q, τ〉 and 〈Coll(ω,X), f〉are equivalent;

3. distinct set mappings on X without free triples yield inequivalent balancedpairs.


Proof. For (1), it is clear that Coll(ω,X) f ∈ P is a condition. Let V [G0], V [G1]be mutually generic extensions of V , and let p0 ∈ V [G0], p1 ∈ V [G1] be condi-tions containing f as a subset. To find a lower bound of p0, p1 in V [G0, G1],let p be the function defined by d(p) = d(p0) ∪ d(p1), p0 ∪ p1 ⊂ p, and if{x0, x1} ⊂ d(p) is a pair such that x0 ∈ d(p0) \ V and x1 ∈ d(p1) \ V thenp(x0, x1) = d(p) \ {x0, x1}. We must show that p is a common lower bound ofp0, p1.

First of all, p0∪p1 is indeed a function, since any pair a in dom(p0)∩dom(p1)is already in the ground model V by the product forcing theorem, and thenp0(a) = p1(a) = f(a). Second, we must verify that p has no free triple. Letb ⊂ d(p) be a triple. If b ⊂ d(p0) or b ⊂ d(p1) holds, then b is not free becausep0, p1 contain no free triples. Otherwise, b can be listed as {x0, x1, x2} such thatx0 ∈ d(p0) \V and x1 ∈ d(p1) \V ; then x2 ∈ p(x0, x1) by the definition of p andb is not free either.

For (2), let 〈Q, τ〉 be a balanced pair. Strengthening Q and τ if necessary,we may assume that Q V ∩X ⊂ d(τ) holds. By a balance argument, for anypoints x0, x1, y ∈ X in the ground model, Q y ∈ τ(x0, x1) or Q y /∈ τ(x0, x1)holds. Let f : [X]2 → P(X) be the function defined by y ∈ f(x0, x1) if theformer alternative in the previous sentence prevails. It is clear that f containsno free triple since τ is forced to contain none. Thus, it will be enough to showthat Q f ⊂ τ .

Suppose towards a contradiction that this fails. Then there must be a con-dition q ∈ Q and points x0, x1 ∈ X such that q τ(x0, x1) 6= f(x0, x1).

8.5. SET MAPPINGS 185

Let G0, G1 ⊂ Q be mutually generic filters containing the condition q andlet p0 = τ/G0, p1 = τ/G1. By the initial choice of the function f , theremust be points y0, y1 ∈ X \ V such that y0 ∈ p0(x0, x1) and y1 ∈ p1(x0, x1).By the product forcing theorem, y0 /∈ V [G1] and y1 /∈ V [G0] holds. Thus,p0(x0, x1) 6= p1(x0, x1) and p0, p1 are incompatible, contradicting the balance ofthe pair 〈Q, τ〉.

(3) is obvious. For the last sentence, suppose that p ∈ P is a condition.Consider the set mapping f : [X]2 → P(X) defined by f(x0, x1) = p(x0, x1) ifboth x0, x1 ∈ X belong to d(p), and f(x0, x1) = X \ {x0, x1} otherwise. It isnot difficult to check that f is a set mapping without free triples. By item (1),f represents a balanced virtual condition below p.

Definition 8.5.4. Let X be an uncountable Polish space. A thin set mappingforcing is a poset P of all functions p such that for some countable set d(p) ⊂ X,p : [d(p)]3 → [d(p)]<ℵ0 is a set mapping without a free quadruple. The orderingis defined by q ≤ p if d(p) ⊂ d(q) and p ⊂ q.

Theorem 8.5.5. Let P be the thin set mapping forcing.

1. If f : [X]3 → [X]<ℵ0 is any set mapping without free quadruples, then〈Coll(ω,X), f〉 is a balanced pair;

2. if 〈Q, τ〉 is any balanced pair, there is a set mapping f : [X]3 → [X]<ℵ0

without free quadruples such that the balanced pairs 〈Q, τ〉 and 〈Coll(ω,X), f〉are equivalent;

3. distinct set mappings on X without free quadruples yield inequivalent bal-anced pairs.

In particular, the poset P is balanced if and only if 2ℵ0 ≤ ℵ2.

Proof. For (1), it is clear that Coll(ω,X) f ∈ p holds. For the balance,suppose that V [G0], V [G1] are mutually generic extensions and p0 ∈ V [G0],p1 ∈ V [G1] are conditions extending f . To construct the lower bound p ofp0, p1, in the model V [G0, G1] choose an enumeration {zn : n ∈ ω} of d(p0) ∪d(p1). Let p be the function on [d(p0) ∪ d(p1)]3 defined by p0, p1 ⊂ p and ifx0, x1, x2 ∈ d(p0) ∪ d(p1) are pairwise distinct points such that x0 ∈ d(p0) \ Vand x1 ∈ d(p1) \ V , then p(x0, x1, x2) is the set of those points of d(p0) ∪ d(p1)which are enumerated before one of the points x0, x1, x2. We must show that pis a lower bound of p0, p1.

First of all, p0∪p1 is indeed a function, since any pair a in dom(p0)∩dom(p1)is already in the ground model V by the product forcing theorem, and thenp0(a) = p1(a) = f(a). Second, we must verify that p has no free quadruple.Let b ⊂ d(p) be a quadruple. If b ⊂ d(p0) or b ⊂ d(p1) holds, then b is notfree because p0, p1 contain no free qudruples. Otherwise, b can be listed as{x0, x1, x2, x3} such that x0 ∈ d(p0) \ V , x1 ∈ d(p1) \ V and x2 is enumeratedafter x3; then x3 ∈ p(x0, x1, x2) by the definition of p and b is not free either.


The proof of (2) is literally copied from the proof of Theorem 8.5.3(2). (3)is obvious. For the last sentence, use an old result of Kuratowski and Sierpinski[34]: for every number n ≥ 2, 2ℵ0 ≤ ℵn−1 holds if and only if there is a setmapping f : [2ω]n → [2ω]<ℵ0 without a free set of size n+ 1. Thus, if 2ℵ0 > ℵ2,by (2) there are no balanced conditions and P is not balanced. On the otherhand, suppose that 2ℵ0 ≤ ℵ2 holds, and let p ∈ P be a condition. To producea balanced virtual condition stronger than p, let g : [X]3 → [X]<ℵ0 be a setmapping without a free set of size 4. Enumerate the set d(p) by 〈xi : i ∈ ω〉and let f : [X]3 → [X]<ℵ0 be a function defined by f(a) = p(a) if a ⊂ d(p),f(a) = g(a) if a ∩ d(p) = 0, and f(a) = g(a) ∪ {xi : i ∈ n} where n is thesmallest number such that xn ∈ dom(a), for sets a such that a 6⊂ d(p) anda ∩ d(p) 6= 0. It is not difficult to see that f is a set mapping without a free setof size 4 such that p ⊂ f ; therefore, by (1) it represents a balanced conditionstronger than p.

8.6 Saturated models on quotient spaces

One can use quotient simplicial complex forcings to add structures to the variousquotient spaces as long as the structures satisfy a well-known amalgamationproperty from model theory. We will first review the definitions. For first orderstructures M,N in the same language, a map π : M → N is an morphism ifit is an injection from dom(M) to dom(N) which transports all relations andfunctions of M to the corresponding relations and functions in N .

Definition 8.6.1. [30] A Fraisse class is a class F of finite structures in a fixedfinite first order language such that

1. F is closed under isomorphism and under induced substructures;

2. (joint embedding property) whenever N0, N1 ∈ F are structures then thereis M ∈ F containing both N0, N1 as induced substructures;

3. (amalgamation) whenever M,N0, N1 ∈ F are structures and π0 : M → N0

and π1 : M → N1 are morphisms, then there is a structure K ∈ F andmorphisms χ0 : N0 → K and χ1 : N1 → K such that χ0 ◦ π0 = χ1 ◦ π1.

We will tacitly assume that our Fraisse classes contain arbitrarily large finitestructures. Fraisse classes are prominent in model theory [41, Theorems 5.3.3and 5.3.5], Ramsey theory [71, 23], and topological dynamics [53, 106]. Theclasses we can handle satisfy a well-known stronger version of amalgamation:

Definition 8.6.2. A Fraisse class F satisfies strong amalgamation if for when-ever M,N0, N1 ∈ F are structures and π0 : M → N0 and π1 : M → N1 aremorphisms, then there is a structure K ∈ F and morphisms χ0 : N0 → K andχ1 : N1 → K such that χ0 ◦ π0 = χ1 ◦ π1, and rng(χ0) ∩ rng(χ1) = (χ0 ◦ π)′′M .

Definition 8.6.3. Let F be a Fraisse class in a finite relational language.

8.6. SATURATED MODELS ON QUOTIENT SPACES 187

1. An F-structure is a structure M in the same language as F such that forevery finite set a ⊂ dom(M), M � a ∈ F ;

2. Let E be a Borel equivalence relation on a Polish space X. The E,F-Fraisse poset P consists of conditions p where p is an F-structure whosedomain is a countable subset of the quotient space X/E. The ordering isdefined by q ≤ p if dom(p) ⊂ dom(q) and p = q � dom(p).

It is obvious that the poset P introduces a F-structure on the space X/E,denoted by Mgen. It is not difficult, but at the same time also not particularlynatural, to describe the poset P (F , E) as a quotient simplicial complex poset:the vertices of the simplicial complex K will be finite F-structures on the E-quotient space, and a finite set a of vertices belongs to K if all structures in aare induced substructures of a single finite F-structure on the E-quotient space.It is obvious that the posets P (F , E) and PK are naturally co-dense; however,we will never use the simplicial complex presentation of P (F , E). To describethe balanced conditions in the poset P (F , E), suppose that M is an F-structureon the virtual E-quotient space X∗∗. Let τM be a Coll(ω,iω1

)〉 name for thecondition π′′M where π is the map from the virtual space (X∗∗)V to the E-quotient space of the extension which maps each virtual class to its realization.Theorem 2.5.6 provides the necessary assurance that the map π is well-definedand its range is a countable set.

Theorem 8.6.4. Let F be a Fraisse class in a finite relational language whichsatisfies strong amalgamation. Let E be a Borel equivalence relation on a Polishspace X. Then in the E,F-Fraisse poset P ,

1. for every F-structure M on X∗∗, the pair 〈Coll(ω,iω1), τM 〉 is balanced;

2. for every balanced pair 〈Q, τ〉 there is a F-structure M on X∗∗ such thatthe pairs 〈Coll(ω,iω1), τM 〉 and 〈Q, τ〉 are equivalent;

3. distinct F-structures on X∗∗ yield inequivalent balanced conditions.

In particular, P is balanced.

Proof. For (1), suppose that M is a F-structure on X∗∗. Let R0, R1 be arbitraryposets and σ0, σ1 are respective R0×Coll(ω,iω1

)- and R1×Coll(ω,iω1)-names

for elements of P such that in the respective posets σ0 ≤ τM and σ1 ≤ τ isforced. Let H0, H1 be mutually generic filters on the respective posets and letp = τM/H0, p0 = σ0/H0 and p1 = σ1/H1; we need to show that p0, p1 arecompatible in the poset P .

For this, first note that by the mutual genericity, dom(p0) ∩ dom(p1) =dom(p). By the assumption on the names σ0, σ1 it is the case that p0 � dom(p) =p1 � dom(p) = p. Write c = dom(p0) ∪ dom(p1). By the strong amalgamationproperty of the Fraisse class F , for each finite set a ⊂ c there is a structureNa ∈ F such that dom(Na) = a and writing a0 = a∩dom(p0) and a1∩dom(p1),Na � a0 = p0 � a0 and N1 � a1 = p1 � a1 both hold. Let U be an ultrafilter on


[c]<ℵ0 such that for each finite set a ⊂ c the set {b : a ⊂ b} is in the ultrafilterU . Let N be the structure on c which is the U -integral of the structures Na. Itis immediate that N � dom(p0) = p0 and N � p1 = p1; the closure of the Fraisseclass under substructures shows that N is a F-structure. It is the desired lowerbound of the conditions p0, p1.

For (2), note that for each relation R in the language of the Fraisse classF and every tuple a of virtual E-classes, it must be the case that either Q τ a belongs to the relation R in the generic structure Mgen, or Q τ a

does not to the relation R in the generic structure Mgen by the balance of thepair 〈Q, τ〉. Let M be the structure on the virtual E-quotient space consistingof those tuples for which the former alternative occurs. It is immediate thatQ × Coll(ω,iω1

) τ ≤ τM in the separative quotient of the poset P . Theequivalence of the two pairs follows from Proposition 5.2.4.

Now, (3) is obvious. For the last sentence, suppose that p ∈ P is a condition.For each finite set a ⊂ X∗∗, choose a structure Na ∈ F such that dom(Na) = aand Na � (dom(p) ∩ a) = p � (dom(p) ∩ a). Let U be an ultrafilter on [X∗∗]<ℵ0

such that for each finite set a ⊂ X∗∗ the set {b : a ⊂ b} is in the ultrafilter U .Let N be the structure on X∗∗ which is the U -integral of the structures Na.It is immediate that N � dom(p) = p; the closure of the Fraisse class undersubstructures shows that N is a F-structure. The pair 〈Coll(ω,iω1

), τN 〉 is abalanced virtual condition below p.

Example 8.6.5. Let E be a Borel equivalence relation on a Polish space X.Let P be the poset of all linear orderings on countable subsets of the E-quotientspace. The poset is designed to add a linear ordering on the quotient space.Since the Fraisse class of finite linear orderings has the strong amalgamationproperty, the poset P is balanced, and its balanced conditions are classified bylinear orders on the virtual E-quotient space.

Example 8.6.6. Let E be a Borel equivalence relation on a Polish space X. LetP be the poset of all tournaments on countable subsets of the E-quotient space.The poset is designed to add a tournament on the quotient space. Since theFraisse class of finite tournaments has the strong amalgamation property, theposet P is balanced, and its balanced conditions are classified by tournamentson the virtual E-quotient space.

The Fraisse posets can be viewed as a particularly well-behaved subclass of asignificantly larger class obtained from standard amalgamation constructions inmodel theory.

Definition 8.6.7. Let T be a complete first-order theory in countable language,with infinite models. Let E be a Borel equivalence relation on a Polish space X.The poset PTE consists of structures p satisfying the theory T whose domain isa countable subset of the E-quotient space. The ordering is that of elementarysubstructure: q ≤ p if the domain of p is a subset of the domain of q, and p isan elementary substructure of q.

8.6. SATURATED MODELS ON QUOTIENT SPACES 189

It is immediate that the union of the structures in the generic filter is a count-ably saturated model of the theory T on the E-quotient space. To classify thebalanced virtual conditions, we use the following definition.

Definition 8.6.8. A balanced theory is a complete consistent theory S in thelanguage of T plus a constant for each virtual E-class which contains T andasserts than no elements other than the virtual E-class constants are algebraicover those constants.

Suppose that S is a balanced theory. Let τS be the Coll(ω,iω1)-name for theset of all countable models p ∈ PTE such that dom(p) contains the set p0 ofrealizations of all ground model virtual E-classes and p |= S. It is not difficultto see that τS is a name for an analytic, nonempty, and open subset of the posetPTE .

Theorem 8.6.9. Let T be a complete first-order theory in countable language,with infinite models. Let E be a Borel equivalence relation on a Polish space X.

1. For every balanced theory S, the pair 〈Coll(ω,iω1), τS〉 is balanced in PTE;

2. for every balanced pair 〈Q, τ〉 there is a balanced theory S such that thebalanced pairs 〈Q, τ〉 and 〈Coll(ω,iω1), τS〉 are equivalent;

3. distinct balanced theories yield inequivalent balanced pairs.

In particular, the poset PTE is balanced.

Proof. For (1), suppose that S is a balanced theory, V [H0], V [H1] are mutuallygeneric extensions of the ground model, and p0, p1 ∈ PTE are conditions in therespective models in the set given by the name τS . We must show that there isa model q ∈ PTE in which both models p0, p1 of T are elementary. Note thatdom(p0) ∩ dom(p1) is exactly the set of all realizations of ground model virtualE-classes by the mutual genericity assumption. Use the algebraicity clause ofDefinition 8.6.8 and a textbook amalgamation theorem [41, Theorem 5.3.5] toconclude that in V [H0, H1] there is a countable model q of T and elementaryembeddings j0 : p0 → q and j1 : p1 → q such that j0, j1 coincide on the set r ofrealizations of virtual E-classes from the ground model, and the sets j′′0 (p0 \ r)and j′′1 (p1 \ r) are disjoint. It is easy to see that such a model q can be realizedin such a way that its domain is a set of E-classes and the embeddings j0, j1 areboth identity maps. Then q is the sought lower bound of the conditions p0, p1.

For (2), let 〈Q, σ〉 be a balanced pair. Strengthening the name σ and theposet Q if necessary, we may assume that σ is in fact a name for a singleelement of PTE as opposed to an analytic subset of PTE and that σ is forcedto contain the realization of every virtual E-class in the ground model. By abalance argument, for every formula φ(~x) and every tuple ~c of virtual E-classesof the same length as ~x, it must be the case that either Q σ |= φ(~c) orQ σ |= ¬φ(~c). Let S be the set of all formulas and tuples of virtual E-classesφ(~c) for which the former option prevails. We need to show that S is a balancedtheory and the balanced pairs 〈Q, τ〉 and 〈Coll(ω,iω1

), τS〉 are equivalent.


Claim 8.6.10. Q forces that no element of σ which is not a realization of avirtual E-class is algebraic over a tuple of virtual E-classes from the groundmodel.

Proof. Suppose towards a contradiction that this fails, and let q ∈ Q be acondition, let η be a Q-name for an element of σ which is not a realization of avirtual E-class, and let φ be a formula, ~c a tuple of virtual E-classes and n ∈ ωbe a natural number such that q σ |= there are precisely n many x such thatφ(x,~c) holds, and η is one of them. Let H0, H1 ⊂ Q be mutually generic filtersover the ground model, both containing q, and let p0 = σ/H0 and p1 = σ/H1.We will show that p0, p1 have no lower bound in the poset PTE , reaching acontradiction with the balance assumption on the pair 〈Q, σ〉.

Suppose then that q is such a lower bound. By elementaricity, q must seeexactly n many solutions to φ(x,~c), and all solutions in p0 and in p1 are solutionsin q. Now, p0 already sees n many solutions, and η/H1 is another solution inp1 which does not belong to p0 or even to V [H0]. Thus, q in fact sees at leastn+ 1 many solutions,a contradiction.

It follows that S is a balanced theory, and also that Coll(ω,iω1)×Q σ ∈ τS .

The equivalence of the balanced pairs 〈Q, τ〉 and 〈Coll(ω,iω1), τS〉 then followsfrom Proposition 5.2.4.

(3) is obvious. For the last sentence, suppose that p ∈ PTE is a condition–acountable model of the theory T . By the upwards Lowenheim–Skolem theorem,there is a model M of the theory T on the virtual E-quotient space in which pis an elementary submodel. Let S be the diagram of M and observe that S isa balanced theory and its corresponding balanced condition is below p.

Example 8.6.11. Let T be the theory of dense linear orders without endpoints.The notion of algebraicity is trivial in the theory T , and T has elimination ofquantifiers. It follows that the balanced theories are exactly the linear orderson the set of virtual E-classes. This should be compared to Example 8.6.5.

8.7 Non-DC variations

All of the partial orders exhibited so far are either σ-closed or ℵ0-distributive,and therefore their corresponding extensions of the symmetric Solovay modelsatisfy DC, the Axiom of Dependent Choices. This is normaly viewed as highlydesirable, as DC is a key tool for developing mathematical analysis and descrip-tive set theory as we know them today. However, balanced forcing can be usedto generate extensions of the symmetric Solovay model in which DC fails. Weinclude one striking example.

Definition 8.7.1. Let X be an uncountable Polish space. The finite-countableposet P associated with X consists of pairs p = 〈ap, bp〉 where ap ⊂ X is afinite set, bp ⊂ X is a countable set, and ap ∩ bp = 0. The ordering is that ofcoordinatewise reverse inclusion.

8.8. SIDE CONDITION FORCINGS 191

The finite-countable poset is rather worthless in the ZFC context. The lackof control over the finite part means that it collapses ℵ1. Namely, for everysequence 〈xα : α ∈ ω1〉 of distinct points of X, the set {α ∈ ω1 : 〈{xα}, 0〉 belongsto the generic filter on P} is forced to be cofinal in ω1 of ordertype ω. However,the finite-countable poset provides a particularly easy answer to an old questionof Woodin: is there a Suslin forcing which is not c.c.c. and yet has no perfectantichain? The finite-countable poset is not c.c.c. as it collapses ℵ1. On the otherhand, it does satisfy ℵ2-c.c. in ZFC, and so by a straightforward absolutenessargument cannot contain a perfect antichain. This should be contrasted withthe convoluted answer to Woodin’s question given in [46].

In the ZF+DC context, the finite-countable poset is much better behaved,as is clear from the following theorem. The generic filter will add a partition ofthe space X into two sets, one of which does not contain an infinite countablesubset and the other contains no perfect subset. Such a partition clearly violatesDC.

In order to classify the balanced conditions in the finite-countable poset, weneed a piece of notation. Whenever a ⊂ X is a finite set, let τa be the Coll(ω,X)-name for a condition in the poset P which is the pair 〈a, (X ∩ V ) \ a〉.

Theorem 8.7.2. Let X be a Polish space and P the associated finite-countableposet.

1. For every finite set a ⊂ X, the pair 〈Coll(ω,X), τa〉 is balanced in P ;

2. for every balanced pair 〈Q, τ〉 there is a finite set a ⊂ X such that thebalanced pairs 〈Q, τ〉 and 〈Coll(ω,X), τa〉 are equivalent;

3. distinct finite sets yield inequivalent balanced pairs.

Proof. For (1), suppose that V [H0], V [H1] are mutually generic extensions ofthe ground model V and p0 = 〈a0, b0〉 ∈ V [H0] and p1 = 〈a1, b1〉 ∈ V [H1] areconditions stronger than p = 〈a, (X ∩ V ) \ a. We have to show that p0, p1 arecompatible, which is to say that a0 ∩ b1 = a1 ∩ b0 = 0 holds. Suppose towardsa contradiction that for example x ∈ a0 ∩ b1 is a point. By the product forcingtheorem, x ∈ V [H0] ∩ V [H1] = V . If x ∈ a then x ∈ b1 is impossible as p1 ≤ p,and if x /∈ a then x ∈ a0 is impossible as p0 ≤ p. A contradiction!

For (2), let τ = 〈a, b〉 where a, b are Q-names for disjoint subsets of X.A balance argument show that for each point x ∈ X, either Q x ∈ a orQ x ∈ b. Let a ⊂ X be the set of all points x ∈ X for which the first alternativeprevails. It is immediate that a is a finite set and Q × Coll(ω,X) τ ≤ τa.Proposition 5.2.4 shows then that the balanced pairs 〈Q, τ〉 and 〈Coll(ω,X), τa〉are equivalent, proving (2). (3) is immediate.

8.8 Side condition forcings

Attempts to force uncountable subsets of Polish spaces with a lack of internalstructure often converge to a kind of a side condition poset. Posets of this kind


are not covered by the previous sections. In this section, we outline severalexamples.

Definition 8.8.1. Let X be a Polish space. Let F (X) denote the Effros Borelspace of all closed subsets of X [55, Section 12.C]. Let I ⊂ F (X) be a Borel setsuch that

⋃I = X and I is not a countable union of elements of I. The poset PI

consists of pairs p = 〈ap, bp〉 where ap ⊂ X and bp ⊂ I are countable sets. Theordering on PI is defined by q ≤ p if ap ⊂ aq, bp ⊂ bq, and (

⋃bp) ∩ aq \ ap = 0.

It is immediate that PI is a σ-closed Suslin forcing. It is designed to add anuncountable set which has countable intersection with every element of I. Forthe classification of balanced conditions, whenever a ⊂ X is a set, write τa forthe Coll(ω,X)-name for the pair 〈a, b〉 where b is the name for the set of allground model coded sets in I.

Theorem 8.8.2. Let X be a Polish space and let I ⊂ F (X) be a Borel set suchthat

⋃I = X. Then

1. for every set a ⊂ X, the pair 〈Coll(ω,X), τa〉 is balanced;

2. for every balanced pair 〈Q, σ〉 there is a set a ⊂ X such that the pairs〈Q, σ〉 and 〈Coll(ω,X), τa〉 are equivalent;

3. distinct subsets of X yield nonequivalent balanced pairs.

In particular, the poset PI is balanced.

Proof. Towards (1), unraveling the definitions, it is sufficient show the following.Suppose that R0, R1 are posets and σ0, σ1 are respective names for countablesubsets of X such that R0 σ0∩

⋃(I∩V ) = 0 and R1 σ1∩

⋃(I∩V ) = 0; then

R0×R1 σ0∩⋃

(I∩V [G1]) = 0 and σ1∩⋃

(I∩V [G0]) = 0 where G0, G1 are therespective product names for the generic filters on the 0th and 1st coordinate.

This, however, is immediate. We will show that R0 × R1 σ0 ∩⋃

(I ∩V [G1]) = 0; the proof of the other assertion is symmetric. Suppose towards acontradiction that η is an R0-name for an element of σ0, χ is an R1-name foran element of I, and 〈r0, r1〉 ∈ R0 ×R1 is a condition forcing that η ∈ χ holds.Let M be a countable elementary submodel of a large structure containing allobjects named so far, let g1 ⊂ R1 ∩M be a filter generic over M containing thecondition r1, and let F = χ/g1. Since F is a closed set in I ∩ V , there mustbe a basic open set O ⊂ X and a condition r′0 ≤ r0 such that O ∩ F = 0 andr′0 R0

η ∈ O. By the genericity of the filter g1 and the forcing theorem appliedin the model M , there must be a condition r′1 ≤ r1 such that r′1 R1 O∩χ = 0.Clearly, 〈r′0, r′1〉 η /∈ χ, contradicting the initial choice of 〈r0, r1〉.

Towards (2), let 〈Q, σ〉 be a balanced pair; enlarging the poset Q if necessarywe may assume that Q collapses the size of the ground model continuum to ℵ0;strengthening the condition σ we may assume that bσ is forced to contain allground model coded elements of I. By a balance argument, for every pointx ∈ X it is the case that either Q x ∈ aσ or Q x /∈ aσ. Let a ⊂ X be the

8.8. SIDE CONDITION FORCINGS 193

set of all those elements of X for which the former alternative prevails. We willshow that Q × Coll(ω,X) σ ≤ τa, which implies the equivalence of σ and τaby Proposition 5.2.4.

To prove the inequality σ ≤ τa, it is enough to show that Q (aσ \ V ) ∩(⋃

(I ∩ V )) = 0. Suppose towards a contradiction that this fails, and let q ∈ Qbe a condition, η be a Q-name for an element of aσ \V and F ∈ I∩V be a closedset such that q η ∈ F . Let G0, G1 ⊂ Q be mutually generic filters containingthe condition q. By a mutual genericity argument, the point η/G0 does notbelong to aσ/G1

, while it does belong to F and F ∈ bσ/G1. This means the

conditions σ/G0, σ/G1 are incompatible in P , violating the balance assumptionon 〈Q, σ〉.

Finally, (3) is immediate from the assumption that X =⋃I.

Example 8.8.3. Let X = 2ω, let Γ be the graph connecting points x, y ∈ X ifthe least number such that x(n) 6= y(n) is even. Let I be the collection of closedΓ-cliques and closed Γ-anticliques. The poset PI adds a generic uncountable setsuch that every Γ-clique or anticlique is countable. This exhibits a balancedextension in which OCA fails, cf. Example 12.2.16.

Example 8.8.4. Let X = ωω and let I be the collection {x ≤ y : x ∈ X} forall functions y ∈ ωω, where ≤ is the everywhere domination ordering on ωω.The poset PI adds a dominating subset of ωω which has only countably manyelements below any given function y ∈ ωω. A set like this must fail to have theBaire property. In a rather different language, Shelah [81] showed that in thePI -extension of the Solovay model every set is Lebesgue measurable, showingthat ZF+DC plus the statement “every set of reals is Lebesgue measurable”does not imply that every set has the Baire property.

In the case when I is the collection of closed nowhere dense subsets of X,there is a more sophisticated variation of the poset PI . For a natural numbern ∈ ω, a set s ⊂ (2ω)n, and a partial function h : n → 2ω write sh = {g ∈(2ω)n\dom(h) : h ∪ g ∈ s}.

Definition 8.8.5. The Lusin poset P is the partial order of all pairs p = 〈ap, bp〉where ap ⊂ 2ω is a countable E0-invariant set and bp is a countable set of pairs〈s, a〉 where s ⊂ (2ω)n is a closed nowhere dense subset of (2ω)n for some n ∈ ω,a ⊂ ap is an E0-invariant set, and for each partial function h : n→ ap \ a whoserange consists of pairwise non-E0-related elements, if dom(h) = n then h /∈ s,and if dom(h) 6= n then the set sh is nowhere dense in the space (2ω)n\dom(h).The order is that of coordinatewise reverse inclusion.

It is not difficult to check that P is a σ-closed Suslin partial ordering. Theposet adds an E0-invariant Lusin subset of 2ω–an uncountable set which inter-sects every nowhere dense set in a countable set. The generic set has a similarproperty in all finite powers though, and in Example 12.2.17 it serves an evenmore refined purpose. The balance of P is an immediate consequence of thefollowing classification theorem. For any E0-invariant set a ⊂ 2ω, let τa be theColl(ω, 2ω)-name for the condition 〈a, b〉 ∈ P where b is the set of all pairs 〈s, a〉


where s is a ground model coded closed nowhere dense subset of (2ω)n in theground model for some n ∈ ω.

Theorem 8.8.6. Let P be the Lusin poset.

1. For every E0-invariant set a ⊂ X, the pair 〈Coll(ω,X), τa〉 is balanced;

2. for every balanced pair 〈Q, σ〉 there is a set a ⊂ X such that the pairs〈Q, σ〉 and 〈Coll(ω,X), τa〉 are equivalent;

3. distinct subsets of X yield nonequivalent balanced pairs.


Proof. Write S = Coll(ω, 2ω). For (1), let R0, R1 be posets and σ0, σ1 be R0×Sand R1×S-names respectively such that R0×S σ0 ≤ τa and R1×S σ1 ≤ τa.Let G0 ⊂ R0 × S and G1 ⊂ R1 × S be mutually generic filters; we must showthat the conditions p0 = σ0/G0, p1 = σ1/G1 ∈ P are compatible.

To this end, write p0 = 〈a0, b0〉 and p1 = 〈a1, b1〉. Let n ∈ ω, let s ∈ (2ω)n

be a nowhere dense set, let 〈a, s〉 ∈ b0 ∪ b1 be a pair, and let h : n→ a0 ∪ a1 bea partial function whose range consists of pairwise non-E0-related elements of(a0∪a1)\a. We must show that either (if dom(h) = n) h /∈ s, or (if dom(h) 6= n)the set sh is nowhere dense.

For definiteness assume that 〈a, s〉 ∈ b0. Let h0 = h∩V [G0] and h1 = h\h0.Since p0 ∈ P holds, the set sh0

is nowhere dense. By the Kuratowski–Ulamtheorem, the set t = {k ∈ (2ω)dom(h1) : sh0∪k is somewhere dense} is meagerand in the model V [G0]. Now, the function h1 is product-Cohen generic over Vby the definition of the name τa. By a mutual genericity argument, h1 is alsoproduct-Cohen generic over V [G0] and therefore does not belong to the set t.Thus the set sh0∪h1

is nowhere dense as required.For (2), suppose that 〈Q, σ〉 is a balanced pair. Strengthening the poset Q

or the condition σ repeatedly, we may assume that Q 2ω ∩ V is countable,and

• for each x ∈ 2ω in V either x ∈ aσ or for no extension p ≤ σ it is the casethat x ∈ ap;

• for each n ∈ ω and each closed nowhere dense set s ⊂ (2ω)n in the groundmodel, either 〈(aσ ∩ V ), s〉 ∈ bσ or for no extension p ≤ σ it is the casethat 〈(aσ ∩ V ), s〉 ∈ bp;

• for each n ∈ ω and each closed nowhere dense set s ⊂ (2ω)n in the groundmodel there is a such that 〈a, s〉 ∈ bσ.

By a balance argument, for each x ∈ 2ω in the ground model, either Q forces thefirst clause of the first item to prevail for x, or Q forces the second clause of thefirst item to prevail for x. Let a ⊂ 2ω be the set of all points for which the firstoption prevails; we will show that Q × S σ ≤ τa, showing that the balancedpairs 〈Q, σ〉 and 〈Coll(ω, 2ω), τa〉 are equivalent by Proposition 5.2.4. To see the

8.9. WEAKLY BALANCED VARIATIONS 195

inequality, it is only necessary to show that the second clause of the second itemabove is impossible for any closed nowhere dense set s ⊂ (2ω)n in the groundmodel. Suppose towards a contradiction that the second clause occurs. Thiscan only happen if there is a Q-name h such that Q forces h : n → 2ω to be apartial map such that rng(h) consists of pairwise non-E0-equivalent, non-groundmodel elements of aσ such that either (if dom(h) = n) h ∈ s or (if dom(h) 6= n)the set sh contains a nonempty open set. In either case, let G0, G1 ⊂ Q bemutually generic filters, and use the third item to find a set a ∈ V [G0] such that〈a, s〉 ∈ bσ/G0

. Then by mutual genericity the set rng(h/G1) consists of pointswhich are not in a. This means that the conditions σ/G0, σ/G1 are incompatiblein P , contradicting the balance assumption on the pair 〈Q, σ〉.

(3) is immediate. For the last sentence, if p ∈ P is a condition, it is notdifficult to see that S forces p to be compatible with τap . The name for a lowerbound is a balanced virtual condition below p.

8.9 Weakly balanced variations

There are a number of restrictions on the balanced extensions of the Solovaymodel–Corollaries 9.1.2, 9.1.5 or Theorems 14.1.1 or 14.2.1 are good examples.Transcending these restrictions requires reaching for a weakly balanced forcing.The weakly balanced arguments are invariably more complicated than the bal-anced ones, and we never obtain a classification of weakly balanced classes. Weinclude two examples reminiscent of the side condition forcings of Section 8.8.The first poset adds an injection from the E-quotient space to the E0-quotientspace, where E is any give Borel equivalence relation on a Polish space. Fora pinned equivalence relation E this is possible to do with a balanced forcing,see Theorem 6.6.3; in the general case, this is impossible to do with balancedforcing by Theorem 9.1.1.

Definition 8.9.1. Let E be a Borel equivalence relation on a Polish space X.The Lusin collapse forcing PE is the poset of all triples p = 〈ap, bp, fp〉 where〈ap, bp〉 is a condition in the Lusin forcing of Definition 8.8.5, and fp is aninjection from the E-space to the set of E0-classes represented in ap. The orderis that of coordinatewise reverse inclusion.

It is not difficult to see that P is a σ-closed Suslin partial order and if G ⊂ P isa generic filter then f =

⋃p∈G fp is a total injection from the E-quotient space

to the E0-quotient space. It is instructive to view the poset PE as a natural two-step iteration. The first step is the (balanced) Lusin poset of Definition 8.8.5adding a subset A of the E0-quotient space. In the second step, an injection ofthe E-quotient space into A is added by straightforward countable approxima-tions. The projection from PE into the Lusin poset is given by the projectionof conditions in p into the first two coordinates.

Theorem 8.9.2. Let E be a Borel equivalence relation on a Polish space X.The Lusin collapse forcing of |E| to |E0| is weakly balanced.


Proof. Write P = PE . Let p ∈ P be a condition; we must find a virtual weaklybalanced condition p ≤ p. Let p = 〈Coll(ω, 2ω), 〈ap, b, fp〉〉 where b is the setof all pairs 〈ap, s〉 where s ⊂ (2ω)n is a closed nowhere dense set coded in theground model for some n ∈ ω. It is immediate that p is a virtual condition; itwill be enough to show that p is weakly balanced.

Before we proceed, we must fix some notation and terminology. Let V [G]be a generic extension of V , and in V [G] let f be a function from the virtualE-quotient space to the quotient E0-space. If M is an intermediate model ofZF between V and V [G], write f |M for the following set. Let γ be the smallestordinal such that in M , every E-pin is equivalent to an E-pin of set-theoreticrank smaller than γ; such an ordinal exists since by Theorem 2.5.6 there are onlyset many virtual E-classes in V [G]. Let f |M be the set of all pairs 〈〈Q, τ〉, y〉such that 〈Q, τ〉 ∈ M is an E-pin of rank smaller than γ, y ∈ 2ω, and forsome virtual E-class c ∈ dom(f), y ∈ f(c) and 〈Q, τ〉 ∈ c. The transfiniteanalysis of f is the sequence of models Mα of ZF given by the recursive formulaMα = V (〈f |Mβ : β ∈ α〉); note that always M0 = V and the models Mα form aninclusion-increasing sequence. The function f has a heart if there is an ordinalα such that

• for every ordinal β ∈ α and every E-pin 〈Q, τ〉 ∈ Mβ there is a classc ∈ dom(f) with 〈Q, τ〉 ∈ c;

• for every virtual E-class c ∈ dom(f), if c∩Mα 6= 0 then c∩⋃β∈αMβ 6= 0.

Note that once such an ordinal α is reached then Mδ = Mα for all ordinalsδ ≥ α. If the function f has a heart, then the least ordinal α as above is itsdepth. The heart of f is the pair 〈M,h〉 where M = Mα and h = f |M .

We apply the above transfinite analysis also to conditions in P . Any E-classc will be identified with the virtual E-class of all E-pins 〈Q, τ〉 such that forsome (all) x ∈ c, Q τ E x; thus, a partial function on the E-quotient spaceis viewed as a partial function on the virtual E-quotient space. The heart of acondition in P is the heart of its last coordinate. The following two key claimscontrol the behavior of hearts of conditions below p.

Claim 8.9.3. Every condition q ∈ P can be strengthened in some generic ex-tension to a condition r ≤ q which has a heart.

Proof. Let V [G] be a generic extension of V , and let q ∈ V [G] be a conditionin P . Work in V [G]. By transfinite recursion on β ≤ ω1 define a finite supportiteration 〈Rβ : β ≤ ω1, Qβ : β < ω1〉 of c.c.c. forcings and Rβ-names τβ so thatRβ forces the following:

• τ0 = 0, γ ∈ β implies τγ ⊂ τβ , and if β is limit then τβ =⋃γ∈β τγ ;

• τβ is a partial function from the virtual E-quotient space to the E0-quotient space which has a heart 〈Mβ , τβ |Mβ〉 of depth β and dom(τβ)contains exactly those virtual E-classes represented in dom(τβ |Mβ);


• Qβ is the finite support product of copies of Cohen forcing on 2ω indexed

by all the virtual E-classes in Mβ which are not represented in⋃γ∈β Mγ

and do not belong to dom(fq), and τβ+1 is the function τβ together withthe function sending each virtual E-class in Mβ as above to the E0-classof the corresponding Cohen real, and sending every virtual E-class in Mβ

which is dom(fq) to the E0-class indicated by fq.

Write R = Rω1 . Let H ⊂ R be a filter generic over V [G]; thus, the models V [G]and V [G][H] have the same ω1. In V [G][H], write fβ = τβ/H for all β ≤ ω1

and f = fω1. Consider the transfinite analysis 〈Mβ : β ∈ ω1〉 of f , starting with

M0 = V . Since the function fq is countable, a counting argument shows thatthere has to be an ordinal β ∈ ω1 such that every virtual E-class representedin Mβ+1 and in dom(fq) is already represented in Mβ . Thus, the transfiniteanalysis of the function fq ∪ fβ equals to 〈Mγ : γ ≤ α〉 and 〈Mβ , fβ |Mβ〉 is itsheart. Consider the triple 〈aq∪rng(fβ), bq, fq∪fβ〉. It is not difficult to see thatit is a virtual condition in P stronger than q. In a suitable generic extension, itturns into a condition r ≤ q with heart 〈Mβ , fβ |Mβ〉.

Claim 8.9.4. Let r ≤ p be a condition in P in some generic extension withheart 〈M,h〉. Then

1. the theory of the model M with parameters in V and the parameter hdepends only on the depth of r;

2. every real in M belongs to some V [b] where b is a finite subset of⋃

rng(h).

Proof. Fix an ordinal α. It turns out that the hearts of conditions ≤ p of depthα are all generated in the same way. Work in V . By transfinite recursion onβ ≤ α define a finite support iteration 〈Rβ : β ≤ α, Qβ : β < α〉 of c.c.c. forcingsand Rβ-names τβ so that Rβ forces the following:

• τ0 = 0, γ ∈ β implies τγ ⊂ τβ , and if β is limit then τβ =⋃γ∈β τγ ;

• τβ is a partial function from the virtual E-quotient space to the E0-quotient space which has a heart 〈Mβ , τβ |Mβ〉 of depth β and dom(τβ)contains exactly those virtual E-classes represented in dom(τβ |Mβ);

• if β = 0 then Q0 is the finite support product of copies of Cohen forcingon 2ω indexed by all the virtual E-classes in M0 = V which do not belongto dom(fp), and τ1 is the function fp together with the function sendingeach virtual E-class in V as above to the E0-class of the correspondingCohen real;

• if β > 0 then Qβ is the finite support product of copies of Cohen forcing on2ω indexed by all the virtual E-classes in Mβ which are not represented in⋃γ∈β Mγ , and τβ+1 is the function τβ together with the function sending

each virtual E-class in Mβ as above to the E0-class of the correspondingCohen real.


We will show that if r ≤ p is a condition with heart 〈M,h〉 of depth α, thenin some forcing extension there is a filter H ⊂ Rα generic over V such thatwriting 〈N, k〉 for the heart of τα/H, then 〈M,h〉 = 〈N, k〉. This will prove thefirst item since the theory of 〈M,h〉 (with parameter h and other parameters inthe ground model) is then exactly the collection of those statements which Rαforces to be true in the heart of τα. The second item of the claim ???

The following claim just restates the information from the previous claim in aform that will be useful later.

Claim 8.9.5. Let r ≤ p be a condition in P in some generic extension withheart 〈M,h〉. Then

1. every E-class in dom(fr) is either a realization of a virtual E-class indom(h) or it is not a realization of any virtual E-class in M ;

2. every finite tuple c of pairwise E0-unrelated elements of ar \⋃

rng(h)) isproduct Cohen-generic over M .

Proof. The first item is an immediate consequence of the definition of theheart. For the second item, we use the fact that r ≤ p. Let n = |c|. Everyclosed nowhere dense subset s ⊂ (2ω)n in the model M belongs to some modelV [d] where d is a finite tuple of pairwise E0-unrelated elements of

⋃rng(h) \⋃

rng(fp)) by Claim 8.9.4(2). By the choice of the virtual condition p, the tuplec∪d is product Cohen-generic over V , and by the product forcing theorem c /∈ s.We have just shown that c does not belong to any closed nowhere dense subsetof (2ω)n in the model M ; that is, c is product Cohen-generic over M .

Finally, we are ready to conclude the proof of Theorem 8.9.2. Let Q0, Q1 beposets in V , with names σ0, σ1 for conditions in P stronger than p. We mustfind, in some generic extension, filters G0 ⊂ Q0 and G1 ⊂ Q1 separately genericover the ground model such that the conditions σ0, σ1 are compatible in P . ByClaim 8.9.3, we may assume that Q0 σ0 has heart of depth α0 and Q1 σ1 hasheart of depth α1. For definiteness, assume that α0 ≤ α1. Move to some forcingextension (such as the Q1-extension) where there is a condition r ≤ p with aheart of depth α1. Let Mβ be the models arising in the transfinite analysis ofr, and let N0 = Mα0

, h0 = fr|N0 and N1 = Mα1, h1 = fr|N1. Thus, 〈N1, h1〉 is

a heart of the condition r; note that 〈N0, h0〉 is also a heart of a condition ≤ p,namely the condition obtained from r by restricting the function fr to the setof E-classes which are realizations of the virtual classes in dom(h0).

Note that in the model N0, there exists a poset S0 and a name η0 for a filteron Q0 generic over V such that S0 forces the heart of σ0/η0 to be 〈N0, h0〉. Thisoccurs because this statement is true in the heart of σ0 and the theory of thehearts depends only on the height by Claim 8.9.4(1). Similarly, in the modelN1 there exists a poset S1 and a name η1 for a filter on Q1 generic over V suchthat S1 forces the heart of σ1/η1 to be 〈N1, h1〉. Let H0 ⊂ S0 and H1 ⊂ S1 be


filters mutually generic over the model N1. Let G0 = η0/H0 and G1 = η1/H1;we claim that the conditions σ0/G0 and σ1/G1 are compatible in P as desired.

To see this, write σ0/G0 = 〈a0, b0, f0〉 and σ1/G1 = 〈a1, b1, f1〉. To show thatthese two conditions are compatible, we must argue that 〈a0, b0〉 and 〈a1, b1〉 arecompatible as conditions in the Lusin forcing, and then show that f0 ∪ f1 is aninjection from E-quotient space to the E0-space. The compatibility in the Lusinforcing is nearly identical to the proof of Theorem 8.8.6(1) and we omit it. Toshow that f0 ∪ f1 is an injection from the E-quotient space to the E0-quotientspace, suppose that c0 ∈ dom(f0) and c1 ∈ dom(f1) are E-classes and work toprove the equivalence f0(c0) = f1(c1) ↔ c0 = c1. By Claim 8.9.5(1) there aretwo cases:

• c0 is a realization of a virtual E-class d ∈ dom(h0). Then f0(c0) = h0(d) =h1(d) and either c1 = c0 in which case f1(c1) = h1(d) = f0(c0), or c1 6= c0and then f1(c1) 6= h1(d) since the function f1 is an injection.

• c0 is not a realization of a virtual E-class in N0. By a mutual genericityargument then, c0 is not an E-class represented in the model N1[H1];in particular c0 6= c1. By Claim 8.9.5(2), the E0-class f0(c0) consists ofpoints Cohen generic over N0. By a mutual genericity argument it mustbe distinct from the E0-classes in N1[H1]; in particular, f0(c0) 6= f1(c1).

This completes the proof.

The following poset introduced in [63] adds an infinite maximal disjoint family,a task impossible to perform with balanced forcing by Theorem 14.1.1. It yieldsa model of ZF+DC where there is an infinite MAD family and every set of realsis Lebesgue measurable by Example 14.3.6.

Definition 8.9.6. The MAD forcing P consists of all pairs p = 〈ap, bp〉 suchthat ap ⊂ [ω]ℵ0 is an infinite countable almost disjoint family, and bp is acountable set of pairs 〈s, a〉 such that s is a partition of ω into finite sets anda ⊂ ap is a countable set. Moreover, for every pair 〈s, a〉 ∈ bp and every finiteset d ⊂ ap \ a, there are infinitely many sets e ∈ s such that

⋃d ∩ e = 0. The

set P is ordered by coordinatewise reverse inclusion.

It is clear that the MAD forcing is Suslin and σ-closed. The union of the firstcoordinates of conditions in the generic filter is forced to be a maximal almostdisjoint family by an elementary density argument.

Theorem 8.9.7. The MAD forcing is weakly balanced.

Proof. Suppose that p ∈ P is a condition, and write b = bp ∪ {〈s, ap〉 : s is aground model partition of ω into finite sets}. It is clear that for any posetQ collapsing the size of the continuum to ℵ0, Q 〈ap, b〉 is a condition in Pstronger than p. It will be enough to show that the pair 〈Q, 〈ap, b〉〉 is weaklybalanced in P .

To this end, suppose that R0, R1 are partial orders and σ0, σ1 are respectiveR0- and R1-names for conditions in P extending 〈ap, b〉; We must produce, in


some generic extension, filters H0 ⊂ R0 and H1 ⊂ R1 separately generic overthe ground model such that the conditions σ0/H0 and σ1/H1 are compatible inP . The following claim is key.

Claim 8.9.8. For every i ∈ 2, the following holds. Whenever r ∈ Ri is acondition and ηj for j ∈ ω are names for elements of aσi \ap, there is a numbern ∈ ω such that for every m ∈ ω there is a strengthening of r forcing

⋃j∈m ηj ∩

[n,m) = 0.

Proof. For definiteness let i = 0. Suppose towards a contradiction that theclaim fails for a condition r ∈ R0 and names ηj for j ∈ m. Then, there is apartition s of ω into finite intervals such that r

⋃j∈b ηj ∩b 6= 0 for every b ∈ s.

This shows that r σ0 6≤ 〈ap, b〉, contradicting the initial assumptions.

Let V [K] be a generic extension collapsing a sufficiently large cardinal andwork in V [K]. An inductive application of the claim makes it possible to findfilters H0 ⊂ R0 and H1 ⊂ R1 separately generic over the ground model, and apartition ω =

⋃j cj of ω into finite sets such that, writing σ0/H0 = 〈a0, b0〉 and

σ1/H1 = 〈a1, b1〉, we have:

• for every x ∈ a0 \ ap, x ⊆⋃{cj : j even} up to finitely many exceptions

and similarly for every x ∈ a1 \ ap, x ⊆⋃{cj : j odd} up to finitely many

exceptions;

• for every infinite collection s ∈ V [H0] of pairwise disjoint subsets of ω,there is e ∈ s which is a subset of some cj for j even, and similarly forevery infinite collection s ∈ V [H1] of pairwise disjoint subsets of ω, thereis e ∈ s which is a subset of some cj for j odd.

We claim that the filters work as required. This is clearly the same as showingthat the pair 〈a0 ∪ a1, b0 ∪ b1〉 belongs to P , since then it is a lower bound ofboth 〈a0, b0〉 and 〈a1, b1〉. First of all, the first item above immediately showsthat a0 ∪ a1 is an almost disjoint family. To confirm the “moreover” demandin the definition of the poset P for the pair 〈a0 ∪ a1, b0 ∪ b1〉, suppose that〈s, a〉 ∈ b0 ∪ b1; for definiteness assume that 〈s, a〉 ∈ b0. Let d ⊂ (a0 ∪ a1) \ a bea finite set; we must produce infinitely many e ∈ s such that

⋃d ∩ e = 0.

Write d0 = d ∩ a0 and d1 = d \ d0. Since the pair 〈a0, b0〉 is a condition inthe poset P , there are infinitely many sets e ∈ s such that

⋃d0 ∩ e = 0. By the

second item above, there must be infinitely many e ∈ s such that⋃d0 ∩ e = 0

and e ⊂ cj for some even number j ∈ ω. By the first item above, there must beinfinitely many sets e among them such that

⋃d1 ∩ e = 0. For all such e ∈ s, it

is the case that⋃d ∩ e = 0. This completes the proof.

The definition of the MAD forcing may seem odd, since it skips the most obviouschoice: the poset Q of all infinite countable almost disjoint families ordered byreverse inclusion. However, the following remains open:

Question 8.9.9. Is the poset Q weakly balanced?

Chapter 9

Preserving cardinalities

9.1 The well-ordered divide

The main feature of the balanced Suslin forcing is that it does not add any well-ordered sequences of elements of the symmetric Solovay model. In particular,the balanced extensions of the Solovay model are barren in the sense of [39, 22].This has a number of cardinality corollaries for the resulting extensions. Thefollowing theorem is stated in terms of Convention 1.7.16.

Theorem 9.1.1. In cofinally balanced extensions of the symmetric Solovaymodel W , every well-ordered sequence of elements of W belongs to W .

Proof. Let P be a Suslin forcing. Let κ be an inaccessible cardinal such that P isbalanced cofinally below κ. Let W be a symmetric Solovay model derived fromκ and work in W . For a formula φ of the language of set theory, element y ∈ 2ω,and a set v ∈ V , write u(φ, y, v) for the unique set x such that W |= φ(x, y, v)if such a unique x exists; otherwise, write u(φ, y, v) = 0. Suppose towards acontradiction that there is a condition p ∈ P , an ordinal α and a P -name τ suchthat p forces τ to be an α-sequence of elements of W which does not belongto W . The name τ is definable from some real parameter z ∈ 2ω and someelements of the ground model. Find an intermediate model V [K] which is anextension of the ground model by a poset of size < κ such that p, z ∈ V [K] andP is balanced in V [K].

Work in V [K]. Let p be a balanced virtual condition in P below p. Sincein W , τ is forced not to belong to W , there must be in V [K] an ordinal β ∈ αand a posets R of size < κ, R-names σ0, σ1 for conditions in P stronger than p,R-names η0, η1 for elements of 2ω, ground models elements v0, v1, and formulasφ0, φ1 such that R Coll(ω,< κ) the following:

• σ0 P τ(β) = u(φ0, η0, v0);

• σ1 P τ(β) = u(φ1, η1, v1);

• u(φ0, η0, v0) 6= u(φ1, η1, v1).

201

202 CHAPTER 9. PRESERVING CARDINALITIES

Working in the model W , let H0, H1 ⊂ R be mutually generic filters, andfor bits b, c ∈ 2 let pbc = σb/Hc ∈ P and ybc = ηb/Hc ∈ 2ω. By the thirditem above, in the model V [K][H0, H1] there must be a bit c ∈ 2 such thatColl(ω,< κ) u(φ0, y00, v0) 6= u(φ1, y1c, v1). The conditions p00, p1c ∈ P arecompatible in W by the balance of the condition p, with a lower bound q ∈P . Since W is the symmetric Solovay extension of the model V [K][H0], theforcing theorem applied with that model yields that q τ(β) = u(φ0, η0, v0).Since W is the symmetric Solovay extension of V [K][H1], the forcing theoremapplied with V [K][H1] yields that q τ(β) = u(φ1, y1c, v1). Finally, since W isthe symmetric Solovay extension of V [K][H0][H1], u(φ0, y00, v0) 6= u(φ1, y1c, v1)holds in W . Thus, the condition q forces two distinct values to the name τ(β),an impossibility.

Corollary 9.1.2. In a cofinally balanced extension of a symmetric Solovaymodel, there is no transfinite uncountable sequences of pairwise distinct Borelsets of bounded rank.

Proof. Since cofinally balanced extensions add no countable sequences of ele-ments of the Solovay model W by Theorem 9.1.1, all Borel sets in W [G] belongto W and have the same Borel rank there as in W [G]. Thus, an uncountablesequence of distinct Borel sets of bounded Borel rank in W [G] would have tobelong to W by Theorem 9.1.1 again. However, there are no such sequences inthe Solovay model W by a result of Stern [87].

Corollary 9.1.3. Let E be a Borel equivalence relation on a Polish space X. Incofinally balanced extensions of the symmetric Solovay model, ℵ1 6≤ |E| holds.

Proof. An ω1-sequence of distinct E-classes would constitute an ω1-sequence ofdistinct Borel sets of Borel rank bounded by the rank of E. Such sequences areruled out by the previous corollary.

Corollary 9.1.3 guarantees that in balanced extensions of the symmetric Solovaymodel, the Friedman–Stanley jump divide is preserved. This follows from ahumble ZF result of independent interest.

Proposition 9.1.4. (ZF) Let X be a set and E an equivalence relation on Xwith all classes countable. If |Xℵ0 | ≤ |E| then |HC| ≤ |E|.

Proof. Let g be an injection from the set of all countable subsets of X to E-classes. Define a function h from the collection of hereditarily countable sets toE-classes by ∈-recursion: h(a) = g(

⋃h′′a). By induction on the minimum of

the rank a and b argue that a 6= b implies that h(a) 6= h(b). Thus the functionh is an injection and the statement of the proposition follows.

Corollary 9.1.5. Let E be a Borel equivalence relation on a Polish space X. Incofinally balanced extensions of a symmetric Solovay model, |E+| 6≤ |E| holds.

9.1. THE WELL-ORDERED DIVIDE 203

Proof. Suppose towards a contradiction that |E+| ≤ |E| holds in the exten-sion. The Proposition yields |HC| ≤ |E|. Clearly, ℵ1 ≤ |HC| holds in ZF,and the concatenation of the cardinal inequalities contradicts the conclusion ofCorollary 9.1.3.

If one wishes to add well-ordered sequences of some objects in the Solovaymodel and maintain control, it is possible to use weakly balanced forcing. Thefollowing theorem, stated in terms of Convention 1.7.16, shows that there willbe no uncountable sequences of reals in the resulting extension.

Theorem 9.1.6. In weakly balanced extensions of the symmetric Solovay modelW , every set of ordinals belongs to W .

Proof. Let κ be an inaccessible cardinal. Let P be a Suslin poset such that Pis weakly balanced below κ. Let W be the symmetric Solovay model derivedfrom κ and work in the model W . Let p ∈ P be a partial order and let τ be aP -name for a set of ordinals; we have to find a strengthening of the condition pwhich decides the membership of every ordinal in the set τ . To this end, notethat both p, τ are definable from a parameter z ∈ 2ω and some parameters inthe ground model.

Let V [K] be an intermediate generic extension obtained by a poset of size< κ such that z ∈ V [K]. Working in the model V [K], let p be a weaklybalanced virtual condition in the poset P , below the condition P . We claimthat for every ordinal α, it is the case that Coll(ω,< κ) p decides in P themembership of the ordinal α in the set τ . Suppose towards a contradiction thatthis is not the case. Then there must be an ordinal α, posets R0, R1 of size < κ,and R0- and R1-names σ0 and σ1 for elements of p stronger than p such thatR0 Coll(ω,< κ) σ0 α /∈ τ and R1 Coll(ω,< κ) σ1 α ∈ τ .

Work in W again. Use the weak balance of p to find filters H0 ⊂ R0 andH1 ⊂ R1 separately generic over V [K] such that the conditions p0 = σ0/H0 andp1 = σ1/H1 are compatible in P . Since W is a symmetric Solovay extensionof both models V [K][H0] and V [K][H1], p0 α /∈ τ and p1α ∈ τ . Thus, thecommon lower bound of p0, p1 must force two contradictory statements, whichis impossible.

It is impossible to strengthen the conclusion of Theorem 9.1.6 to sequences ofelements of the Solovay model. For any Borel equivalence relation E on a Polishspace X, Theorem 8.9.2 produces a weakly balanced extension of the symmetricSolovay model in which |E| ≤ E0. In the natural case E = F2, Proposition 9.1.4applied in the resulting model shows that |HC| ≤ |E0| there, in particular thereis an ω1-sequence of E0-classes. Note that the conclusion of Corollary 9.1.5 failsas well in the model.

As the last remark in this section, many preservation theorems in this bookcan be combined since the various properties of the posets concerned are pre-served under product. In view of the many mutual consistency results concern-ing ZFC such as those contained in [100], one can ask for example whether therecan Σ2

1 sentences φ0 and φ1 such that both ZF+DC+φ0 and ZF+DC+φ1 are


consistent with the statement ψ asserting the nonexistence of an uncountablesequence of pairwise distinct reals, while ZF+DC+φ0+φ1 implies ¬ψ. The fol-lowing two examples provide an affirmative answer, in the second case φ0+φ1

even implies that the reals are well-ordered in ordertype ω1.

Example 9.1.7. Let φ0 be the statement “there is an uncountable sequenceof E0-classes” and let φ1 be the statement “there is an E0-transversal”. Theconjunction ψ ∧ φ0 holds in the abovementioned weakly balanced extension ofthe Solovay model. The conjunction ψ ∧ φ1 holds in the balanced extension ofthe Solovay model adding an E0-transversal with countable approximations byCorollary 9.1.2. Finally, the conjunction φ0 ∧φ1 immediately implies ¬ψ in ZF.This example shows that a product of a balanced and a weakly balanced forcingis not necessarily weakly balanced.

Example 9.1.8. Let φ0 be the statement “there is an acyclic decomposition of2ω” as in Definition ??. Let X = (2ω)ω and let A ⊂ X be the set of all x ∈ Xsuch that Turing reducibility linearly orders rng(x). Let E = F2 � A and let φ1

be the statement “|E| ≤ |2ω|”. The theories ZF+DC+φ0 and ZF+DC+φ1 areseparately consistent with ψ, but the theory ZF+DC+φ0+φ1 implies that thereals are well-ordered of ordertype ω1.

Proof. Let P0 be the acyclic decomposition forcing of Definition ??. It is bal-anced if CH holds by Theorem ??, and so by Theorem 9.1.1 the P0-extension ofthe symmetric Solovay model contains an acyclic decomposition of 2ω while itdoes not contain any uncountable sequence of pairwise distinct reals. Let P1 bethe collapse poset of E to 2ω of Definition 6.6.2. Since the virtual E-classes areclassified by subsets of 2ω linearly ordered by Turing reducibility and each suchset can have size at most ℵ1, it follows that λ(E) = 2ℵ1 and by Corollary 6.6.4the poset P1 is balanced if 2ℵ0 = 2ℵ1 holds. By Theorem 9.1.1, the P1 extensionof the symmetric Solovay model satisfies |E| ≤ |2ω| while it does not containany uncountable sequence of pairwise distinct reals. Note that the posets P0, P1

are balanced under incompatible assumptions additional to ZFC, and as a resultZFC proves that the product P0 × P1 is not balanced.

Now, argue in the theory ZF+DC+φ0+φ1. Let c : [2ω]2 → ω be an acyclicdecomposition, and let f be an injection from the E-quotient space to 2ω. LetM be the model of sets hereditarily ordinally definable from the parametersc, f . As is well-known, M is a model of ZFC. Since c � M ∈ M is an acyclicdecomposition in M , by Proposition ?? M satisfies CH. The argument will becomplete if we show that 2ω ⊂M .

We first show that M contains uncountably many reals. Suppose towardscontradiction that this fails. Working in M , write κ = ωM1 and find an uncount-able sequence 〈xα : α ∈ κ〉 of elements of 2ω which is increasing in the Turingreducibility order. Consider the injective map g : P(κ) ∩M → 2ω defined byg(b) = f({xα : α ∈ b}); note that the definition is enabled by the initial contra-dictory assumption. The map g is definable, therefore belongs to M and in M ,it witnesses 2ℵ1 ≤ 2ℵ0 , in contradiction with the Continuum Hypothesis.

Now, we are ready to show that 2ω ⊂ M . Suppose that x ∈ 2ω is a point.Since M contains uncountably many elements of 2ω, there must be distinct

9.2. THE SMOOTH DIVIDE 205

points y0, y1 ∈ 2ω ∩M such that c(x, y0) = c(x, y1) = n for some number n ∈ ω.There cannot be any other point x′ ∈ 2ω such that c(x′, y0) = c(x′, y1) = n,because the sequence x, y0, x

′, y1 would form a monochromatic cycle. Thus, xis defined by the demand c(x, y0) = c(x, y1) = n and so it is an element of Mas desired.

9.2 The smooth divide

A basic result in the theory of analytic equivalence relations says that there isno Borel reduction from E0 to the identity on 2ω. A basic concern of our theorythen is when |E0| > |2ω| holds in the balanced extensions of the symmetricSolovay model. The take home lesson of this section is that if P is a balancedSuslin poset whose balanced conditions are naturally organized into a compactHausdorff space, then in the P -extension of the symmetric Solovay model, |E0| >|2ω| indeed holds. There are other ways to preserve the inequality |E0| > |2ω|discussed in Chapter 11, but the compactness arguments are the most elegantand cover a lot of ground.

Definition 9.2.1. A Suslin poset P is compactly balanced if there is a definablecompact Hausdorff topology T on the set B of all equivalence classes of balancedvirtual conditions such that

1. for every p ∈ P the set {p ∈ B : p ≤ p} ⊂ B is nonempty and T -closed.

Moreover, if V [H0] ⊂ V [H1] are generic extensions of V then

2. for every balanced virtual condition p0 ∈ V [H0] there is a balanced virtualcondition p1 ∈ V [H1] such that p1 ≤ p0 holds;

3. the relation {〈p0, p1〉 ∈ BV [H0] × BV [H1] : p1 ≤ p0} is closed in TV [H0] ×TV [H1].

The definition seems to be obscure at the first reading, but it is fully justifiedby the natural compact Hausdorff topologies present in the examples. Onerather (intentionally) unclear feature is the definability of the topology T . Itis central in that otherwise the definition would have no content and the keyitem (3) could not be stated without it. The definition is allowed to have realparameters. However, the nature of the definition of T is left undetermined.In the examples, the topology T is typically nonseparable, and the compactHausdorff space 〈B, T 〉 can be in principle very complicated. The followingtheorem is the main result of this section. Its statement uses Convention 1.7.16.

Theorem 9.2.2. In compactly balanced extensions of the symmetric Solovaymodel, |E0| > |2ω| holds.

The proof uses a technical tool which has been studied in its own right: theVitali forcing, also called E0-forcing in [48, Section 10.9] or in [101, Section


4.7.1], where it was isolated for the first time. To define it, consider the σ-idealI on 2ω generated by Borel partial E0-transversals, and let Q be the poset ofBorel I-positive sets ordered by inclusion. As every poset of this form, Q addsa generic point in 2ω–the unique point which belongs to all Borel sets in thegeneric filter [101, Proposition 2.1.2].

Fact 9.2.3. Let Q be the Vitali forcing and let G ⊂ Q be a generic filter. Then

1. [101, Theorem 4.7.3] every real in V [G] is the image of the generic pointxgen under a ground model Borel function (Q is proper);

2. [84, Proposition 4.5] for every set a ⊂ ω in V [G], either a or its comple-ment contain a ground model infinite set (Q adds no independent reals);

3. [103, Theorem 2.11] every real definable in V [G] from [xgen]E0 and groundmodel parameters belongs to the ground model.

Proof of Theorem 9.2.2. Let κ be an inaccessible cardinal. Suppose that P is aSuslin forcing such that P is compactly balanced below κ. Let W be a symmetricSolovay model derived from κ and work in W . Towards a contradiction, assumethat p ∈ P is a condition and τ is a P -name for an injection from the E0-classesto 2ω. Both p, τ are definable from some real parameter z ∈ 2ω. Let V [K] bean intermediate generic extension of V obtained by a forcing of size < κ suchthat z ∈ V [K] holds, and work in the model V [K].

Let Q0 be the poset of infinite subsets of ω, ordered by inclusion. Let Q1

be the Vitali forcing of Borel subsets of 2ω on which E0 is not smooth, orderedby inclusion. Let 〈U, y〉 be an object Q0 × Q1-generic over the model V [K];that is, U is an ultrafilter on ω and y ∈ 2ω is a point. By the σ-closure of Q0,the models V [K] and V [K][U ] have the same reals, in particular they evaluatethe definition of the Vitali forcing in the same way and so y is a Vitali-genericpoint over V [K][U ] by the product forcing theorem. By Fact 9.2.3(2), and agenericity argument, the ultrafilter U still generates an ultrafilter on ω in themodel V [K][U ][y].

Now, work in the model V [K][U ]. Let p0 ≤ p be a balanced virtual conditionbelow p. Let also χ be any Q1-name for a balanced virtual condition in P inthe model V [K][U ][y] below p0. This is possible by (2) of Definition 9.2.1. Notethat the model V [K][U ][y] has more reals than V [K][U ]; thus, the balance ofthe condition p0 does not necessarily transfer from V [K][U ] to V [K][U ][y] andp0 may have to be improved in a nontrivial way to get a balanced conditionin V [K][U ][y]. For each n ∈ ω, let yn ∈ 2ω be the binary sequence obtainedfrom y by replacing its first n entries by 0. Note that for each n ∈ ω, yn isstill Q1-generic over V [K][U ], and V [K][U ][y] = V [K][U ][yn]. In the modelV [K][U ][y], let p1 be the U -limit of the sequence χ/yn for n ∈ ω. By (3) ofDefinition 9.2.1, it is the case that p1 ≤ p0 ≤ p. Note also that the definition ofp1 does not depend on y per se, but only on the E0-class of y. The treatmentdivides into two cases according to the fate of the value τ([y]E0

) as forced by p1,and a contradiction is reached in each case.


Case 1. In the model V [K][U ][y], there is no point u ∈ 2ω such that Coll(ω,<κ) p1 P τ([y]E0) = u. In such a case, there have to be disjoint basicopen subsets O0, O1 of 2ω, posets R0, R1 of size < κ, and respective R0, R1-names σ0, σ1 for conditions in P stronger than p such that R0 Coll(ω,<κ) σ P τ([y]E0

) ∈ O0, and similarly for subscript 1. In the model W , letH0 ⊂ R0 and H1 ⊂ R1 be filters mutually generic over the model V [K][U ][y]and consider the conditions σ0/H0, σ1/H1 ∈ P . By the balance of the conditionp1, the conditions σ0/H0, σ1/H1 are compatible. The forcing theorem applied inboth models V [K][U ][y][H0] and V [K][U ][y][H1] now says that in W , their lowerbound forces τ([y]E0

) to belong simultaneously to O0 and to O1, an impossibility.

Case 2. There is a point u ∈ 2ω in V [K][U ][y] such that Coll(ω,< κ) p1 Pτ([y]E0

) = u. This point u is clearly unique, and since p1 is definable from[y]E0

, so is u. By Fact 9.2.3(3), u ∈ V [K][U ] holds. Working in V [K][U ], letq ∈ Q1 be a condition forcing Coll(ω,< κ) p1 P τ([xgen]E0

) = u, where xgen

is the Q1-name for its generic point. Let H0, H1 ⊂ Q be filters mutually genericover V [K][U ] containing the condition q. Let y0, y1 ∈ 2ω be the respectivegeneric points; observe that y0, y1 are not E0-related by mutual genericity. Writealso p0 = p1/H0 and p1 = p1/H1. These are two virtual conditions in Pstrengthening p0. By the balance of p0, they are compatible in the poset P .Then, in the model W , their lower bound forces that τ([y0]E0

) = τ([y1]E0) = u.

This contradicts the assumption that τ was forced to be an injection.

Example 9.2.4. Let P be the poset of P(ω) modulo finite. By Theorem 7.1.2,the balanced virtual conditions are classified by nonprincipal ultrafilters on ω.Equip the set of balanced classes by the topology of the remainder βω. It isimmediate that the topology witnesses the fact that P is compactly balanced.

Corollary 9.2.5.

1. Let P be the poset P(ω) modulo finite. In the P -extension of the symmetricSolovay model, |E0| > |2ω|;

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds, thereis a Ramsey ultrafilter on ω, and yet |E0| > |2ω|.

Example 9.2.6. Let A be a Ramsey sequence of finite structures and PA theassociated σ-closed poset as in Definition 7.3.1. Theorem 7.3.4 provides a clas-sification of the balanced virtual conditions: they correspond to Ramsey se-quences of ultrafilters on the respective domain sets Dn for n ∈ ω. Sequences ofultrafilters of this type form a closed subset of the product

∏nD∗n of the respec-

tive compact sequences of ultrafilters, and so form a compact Hausdorff space.Claim 7.3.6 confirms the extension property Definition 9.2.1(2). It follows thatthe poset PA is compactly balanced.

The conjunction of Examples 9.2.6 and 7.3.8 now yields the following:

Corollary 9.2.7. Let k ∈ ω be a number.


1. Let P be the poset PA where A is a sequence enumerating all finite orderedgraphs with no clique of size k. In the P -extension of the symmetricSolovay model, |E0| > |2ω|;

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds, thereis an ultrafilter U on ω such that U → (U, k)2 and U 6→ (U, k + 1)2, andyet |E0| > |2ω|.

Example 9.2.8. Let P be the Fin×Fin poset of Definition 7.2.1. The poset Pis compactly balanced.

Proof. The balanced virtual conditions of P were classified in Theorem 7.2.2.They correspond to ultrafilters on ω×ω∗ which do not contain the set n×ω∗ forany number n ∈ ω. Thus, the space of balanced virtual conditions is naturallyorganized into a definable compact Hausdorff space. It is necessary to verify theconditions (2, 3) of Definition 9.2.1. Let V [H0] ⊂ V [H1] be generic extensions.

We first evaluate the complexity of the order between the balanced virtualconditions in V [H0] and V [H1]. Consider the partial map f : (ω × ω∗)V [H1] →(ω × ω∗)V [H0] defined by f(n,U) = 〈n,U ∩ V [H0]〉; if U ∩ V [H0] /∈ V [H0], thefunctional value is left undefined. Unraveling the definitions, a balanced virtualcondition in V [H0] correspoding to an ultrafilter W0 on ω × ω∗ is weaker thanthe balanced virtual condition in V [H1] corresponding to some ultrafilter W1

on ω × ω∗ just in case for every set A ∈ W0, the set f−1A belongs to W1. Inview of the topology on the ultrafilter spaces, this is a closed relation.

To see that every balanced virtual condition in V [H0] can be extended to abalanced virtual condition in V [H1], suppose that W0 is an ultrafilter on ω×ω∗in the model V [H0]. Considering the function f from the previous paragraph, itis obvious that the collection {f−1A : A ∈W0} of subsets of ω×ω∗ in the modelV [H1] has the finite intersection property, and therefore can be extended to anultrafilter W1. The ultrafilter W1 corresponds to a balanced virtual conditionin the model V [H1] stronger than the one corresponding to W0.

Corollary 9.2.9.

1. Let P be the Fin×Fin poset of Definition 7.2.1. In the P -extension of thesymmetric Solovay model, |E0| > |2ω|;

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds, thereis an ultrafilter U on ω such that ???, and yet |E0| > |2ω|.

Question 9.2.10. Is there a Borel filter F on ω such that ZF+DC proves thatexistence of a nonprincipal ultrafilter extending F implies |E0| ≤ |2ω|?

Example 9.2.11. Let F be a Fraisse class in a finite relational language withstrong amalgamation. Let E be a Borel equivalence relation on a Polish space X.Let P (F , E) be the poset of Definition 8.6.3 for adding a F-structure to the E-quotient space. Then P (F , E) is compactly balanced: the balanced conditionsare classified by F-structures on the virtual quotient space X∗∗, which form acompact subspace of P([X∗∗]n) where n corresponds to the arity of the relations


in the language for F . It is easy to verify that the demands of Definition 9.2.1are satisfied with the inherited topology.

Corollary 9.2.12. Let E be a Borel equivalence relation on a Polish space X.

1. Let P be the linearization poset for E of Example 8.6.5. In the P -extensionof the symmetric Solovay model, |E0| > 2ω;

2. it is consistent relative to an inaccessible cardinal that ZF+DC holds, theE-quotient space can be linearly ordered, and yet |E0| > |2ω|.

Another class of examples stems from posets which select a single structure oneach E-class from a compact class of structures, where E is a countable Borelequivalence relation.

Example 9.2.13. Let X be a Polish space and let Γ be a locally finite bipartiteBorel graph on X satisfying the Hall’s marriage condition. Consider the posetP adding a perfect Γ-matching as in Example 6.2.4. The balanced virtualconditions are classified by perfect matchings. The set of perfect matchings isnaturally viewed as a compact subset of P(Γ). It is not difficult to check thatthe inherited compact topology satisfies the demands of Definition 9.2.1.

Corollary 9.2.14. Let X be a Polish space and let Γ be a locally finite bipartiteBorel graph on X satisfying Hall’s marriage condition.

1. Let P be the complete matching poset for Γ. In the P -extension of thesymmetric Solovay model, |E0| > |2ω|;

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds, Γhas a perfect matching, and yet |E0| > |2ω|.

Example 9.2.15. Let X be a Polish space and let Γ be a Borel, acyclic, locallyfinite graph on X in which every vertex has degree at least 2; view Γ as asymmetric subset of X2. Consider the poset P of Example 6.2.5, adding anorientation of Γ in which every vertex has exactly one point in its outflow, orin other words which selects an end to each connected component of Γ. Thebalanced conditions are classified by all such orientations, which form a compactsubset of P(Γ). It is not difficult to check that the inherited compact topologysatisfies the demands of Definition 9.2.1.

This example is somewhat singular in that it is the only compactly balancedposet for which we are able to confirm that it introduces some new cardinalinequalities between quotient cardinals. Namely, writing E for the Γ-path-connectedness equivalence relation, in the P -extension of the symmetric Solovaymodel |E| ≤ |E0| must hold. To see this, consider the equivalence relation Fon the set D ⊂ Xω of all sequences consisting of pairwise E-equivalent points,making two such sequences ~x0, ~x1 F -equivalent if some tail of ~x0 is equal to sometail of ~x1. It is not difficult to see that F is a hypersmooth equivalence relation[48, Theorem 8.3.1] with all classes countable and so by a standard result [48,


Theorem 8.1.1], F is Borel reducible to E0. Consider the P -generic orientation~Γ of the graph Γ; each vertex has exactly one point in its outflow. For eachpoint x ∈ X let h(x) = 〈xi : i ∈ ω〉 where x0 = x and xi+1 is the unique point

in the ~Γ-outflow of xi. Then h is a (non-Borel) reduction of E to F and so inthe P -extension |E| ≤ |F | ≤ |E0| holds.

Corollary 9.2.16. Let X be a Polish space and let Γ be a Borel, acyclic, locallyfinite bipartite Borel graph on X in which every vertex has degree at least 2.

1. Let P be the poset selecting an end to each connected component of Γ. Inthe P -extension of the symmetric Solovay model, |E0| > |2ω|;

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds, thereis a function assigning exactly one end to each connected component of Γ,and yet |E0| > |2ω|.

Example 9.2.17. Let n ∈ ω be a number, let X be a Polish space and let Γ bea locally countable Borel graph on X such that every finite subgraph of Γ haschromatic number ≤ n. Consider the poset P adding a Γ-coloring by n colorsas described in Example 6.2.6. The balanced virtual conditions are classifiedby total Γ-colorings by ≤ n-colors; these form a compact subset of nX . It isnot difficult to see that the inherited compact product topology satisfies thedemands of Definition 9.2.1.

Corollary 9.2.18. Let X be a Polish space and let Γ be a locally countableBorel graph on X such that each finite subset of Γ has chromatic number ≤ nfor some fixed number n ∈ ω.

1. Let P be the coloring poset for Γ. In the P extension of the symmetricSolovay model, |E0| > |2ω|;

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds, Γhas chromatic number ≤ n, and yet |E0| > |2ω|.

Question 9.2.19. Is there a Borel graph G on a Polish space X such that eachfinite subgraph of G has chromatic number ≤ n, and ZF+DC proves that if Ghas finite chromatic number then |E0| ≤ |2ω|?

Example 9.2.20. Let Γ,∆ be abelian Polish groups, with ∆ divisible and com-pact. Let P (Γ,∆) be the poset adding a homomorphism from Γ to ∆ as isolatedin Definition 8.2.1. As proved in Theorem 8.2.2, the balanced conditions areclassified by homomorphisms from Γ to ∆. The space of homomorphisms isa closed subset of ∆Γ equipped with the product topology. It is not difficultto see that the demands of Definition 9.2.1 are met. Let us elaborate on theextension property (2). If h : Γ→ ∆ is a homomorphism in some generic exten-sion V [H0] then it can be extended to a homomorphism in any larger forcingextension V [H1] by the divisibility of ∆ and Baer’s criterion [3]. Note that thegroup ∆ remains abelian and divisible in all generic extensions by Mostowskiabsoluteness.


Corollary 9.2.21. Let Γ,∆ be Polish abelian groups, with Γ uncountable andtorsion free, and ∆ divisible and compact.

1. In the P (Γ,∆) extension of the symmetric Solovay model, |E0| > |2ω|;

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds, thereis a discontinuous homomorphism from Γ to ∆, and yet |E0| > |2ω|.

Proof. Recal from Example 8.2.3 that the generic homomorphism added by theposet P (Γ,∆) is discontinuous. Other than that, the corollary follows fromTheorem 9.2.2.

The assumption that ∆ be compact cannot be dropped entirely from the state-ment of the corollary by the following ZF observation:

Proposition 9.2.22. (ZF+DC) Let Y be a separable Banach space. If there isa discontinuous homomorphism h : Y → Y then there is an E0-transversal, inparticular |E0| ≤ |2ω|.

Proof. The discontinuity of the homomorphism and the DC assumption yielda sequence 〈yn : n ∈ ω〉 of elements of Y such that |yn| > Σm>n|ym| and|h(yn+1) > 2|h(yn)|. Now, for every x ∈ 2ω let g(x) = Σ{yn : x(n) = 1} ∈ Y .Let d ⊂ 2ω be any E0-class. The homomorphism assumptions on h show thatthe function g � d is injective and also, the norms of the points in h◦g(d) divergeto infinity. Thus, d contains a finite subset of points whose h◦g-images have thesmallest possible norm, and one can let xd ∈ d be the lexicographically smallestpoint in d the norm of whose h ◦ g-image is as small as possible. The set {xd : dis an E0-class} is an E0-transversal.

Example 9.2.23. The Kurepa poset P on a Polish space X of Definition 8.4.2 iscompactly balanced. To see this, equip 2X , identified with P(X), with the usualproduct topology, refer to Theorem 8.4.3 to argue that the balanced conditionsare classified by subsets of P(X) closed under intersections and containing X,and observe, that sets of this type form a closed and therefore compact subsetof P(X). The properties (1-3) of Definition 9.2.1 are easily verified for thistopology.

Corollary 9.2.24. Let X be a Polish space.

1. In the Kurepa poset extension of the symmetric Solovay model, |E0| > |2ω|;

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds, thereis a Kurepa family on X, and yet |E0| > |2ω|.

As a final remark in this section, the consistency results obtained here can becombined using the countable support product. The balanced virtual conditionsin a countable product of posets are simply sequences of balanced virtual con-ditions in each coordinate (Theorem 5.2.9). A product of Hausdorff compactspaces is Hasudorff compact again; thus, a product of countably many compactly


balanced forcings with full support is compactly balanced again. However, themachinery of Chapter 11 (which can also be used to show that the smooth divideis preserved in certain extensions) seems to be incompatible with the compactlybalanced approach.

9.3 The turbulent divide

We wish to transfer the ergodicity theorem 3.3.5 to cardinal inequalities ingeneric extensions of the Solovay model. The following variation of balance willbe central in this effort.

Definition 9.3.1. Let P be a Suslin forcing.

1. A pair 〈Q, τ〉 is placid if Q τ ⊂ P is an analytic set and wheneverR0, R1 are posets and in some ambient forcing extension, H0 ⊂ R0 × Qand H1 ⊂ R1×Q are filters separately mutually generic over V such thatV [H0]∩V [H1] = V , then any two conditions p0 ≤ Στ/H0 and p1 ≤ Στ/H1

in the respective models V [H0], V [H1] are compatible in P .

2. The poset P is placid if below every condition p ∈ P there is a virtualcondition p ≤ p which is placid.

As an initial example, consider the poset P of countable functions from 2ω to 2ordered by reverse inclusion. The balanced virtual conditions are classified bytotal functions from 2ω to 2. It turns out that every such a virtual condition p isplacid. If V [H0] and V [H1] are generic extensions such that V [H0]∩V [H1] = Vand p0 ∈ V [H0] and p1 ∈ V [H1] are conditions stronger than p, then p0 ∪p1 is a function since dom(p0) \ V and dom(p1) \ V are disjoint sets and thefunctions p0, p1 agree on the entries from V . Therefore, the conditions p0, p1

are compatible as desired.Placid posets share many preservation properties. The following theorems arestated using the standard Convention 1.7.16. The first theorem in addition usesa standard parlance.

Definition 9.3.2. (ZF) The phrase “the turbulent divide is preserved” denotesthe following statement: Let E be an analytic equivalence relation on a Polishspace induced as an orbit equivalence relation of a turbulent Polish group action.Let F be a virtually placid analytic equivalence relation on a Polish space. Then|E| 6≤ |F |.

Theorem 9.3.3. In cofinally placid extensions of the symmetric Solovay model,the turbulent divide is preserved.

Proof. Let Γ be a Polish group, turbulently acting on a Polish space X, result-ing in the equivalence relation E. Let F be a virtually placid orbit equivalencerelation on a Polish space Y . Let P be a Suslin forcing and let κ be an inac-cessible cardinal such that P is cofinally placid below κ. Let W be a symmetric

9.3. THE TURBULENT DIVIDE 213

Solovay model derived from κ and work in the model W . Suppose towardsa contradiction that there is a condition p ∈ P and a P -name for a functionwhich is an injection from the E-quotient space to the F -quotient space. Thecondition p as well as the name τ must be definable from some ground modelparameters together with a parameter z ∈ 2ω. Use the assumptions to find anintermediate model V [K], which is obtained from V by a poset of cardinalitysmaller than κ, contains z and in which the poset P is placid.

Work in the model V [K]. Let p ≤ p be a virtual condition in P which isweakly placid. Let PX be the poset for adding a Cohen point of the space X.That is, PX is the poset of all nonempty open subsets of X ordered by inclusion,adding a point x ∈ X. There must be a poset R of size < κ and PX ×R-namesσ for an element of the poset P stronger than p and η for an element of Y suchthat PX ×R Coll(ω,< κ) σ P τ([x]E) = [η]F . There are two cases.Case 1. There is a nonempty open set O ⊂ X and a condition r ∈ R suchthat the name η is F -pinned below 〈O, r〉. In this case, let x0, x1 ∈ X andH0, H1 ⊂ R be points in O and filters on R containing r mutually generic overthe model V [K]. Let p0 = σ/x0, H0 ∈ P ∩ V [K][x0][H0] and p1 = σ/x1, H1 ∈P ∩ V [K][x1][H1] and let y0 = η/x0, H0 and y1 = η/x1, H1.

The points x0, x1 ∈ X are mutually Cohen generic, and since their respectiveE-classes are meager, it must be the case that they are unrelated. By the caseassumption, the points y0, y1 ∈ F are F -related. By the balance of the conditionp, the conditions p0, p1 are compatible in the poset P . Now, the forcing theoremapplied in the model V [K] shows that in the model W , the lower bound of p0, p1

forces in P that τ([x0]E) and τ([x1]E) are both equal to [y0]F , contradicting theinjectivity assumption on τ .Case 2. Case 1 fails. This is to say that η is forced not to be a realization ofany F -pinned class of V [K]. Let PΓ be the Cohen poset on the Polish group Γand let x0 ∈ X and γ ∈ Γ be points mutually generic for PX , PΓ over the modelV [K], and write x1 = γ · x0.

Since the action of the group is a continuous open map from Γ × X toX, the point xy is PX -generic over the model V [K] by Proposition 3.1.1 ap-plied in V [K]. By Theorem 3.2.2 (this is the only place where we use theturbulence assumption), V [K][x0] ∩ V [K][x1] = V [K] holds. Let H0, H1 ⊂ Rbe filters mutually generic over the model V [K][γ, x0]. By the mutual gener-icity, V [K][x0][H0] ∩ V [K][x1][H1] = V [K] holds. Write p0 = σ/x0, H0 andp1 = x1, H1, and also y0 = η/x0, H0 and y1 = η/x1, H1.

By their initial choice, the points x0, x1 ∈ X are E-related. By the virtualplacidity assumption on the equivalence relation F , the points y0, y1 ∈ Y arenot F -related: if they were, they would be realizations of some virtual F -classin V [K], contradicting the case assumption. By the placidity assumption on thevirtual condition p, the conditions p0, p1 have a common lower bound. Now, theforcing theorem applied in V [K] shows that in the model W , the lower boundof p0, p1 forces in P that τ([x0]E) must be equal simultaneously to [y0]F and[y1]F . This is a contradiction, as τ is forced to be a function.

Other preservation properties are proved in Section 12.3, respectively in Sec-


tion 12.2 based on the observation (Example 12.2.8) that placid forcings areincluded in the much wider class of Bernstein balanced forcings.

It is now time to provide a long but not exhaustive list of examples withtheir associated corollaries.

Example 9.3.4. Suppose that K is a Borel simplicial complex on a Polish spaceX of Borel coloring number ℵ1. Then the poset PK is placid, and every balancedvirtual condition is placid. To see this, revisit Theorem 6.3.3 and observe thatits proof uses the product forcing theorem only to ascertain that if V [H0], V [H1]are mutually generic extensions of the ground model, then V [H0]∩ V [H1] = V .

Example 9.3.5. Suppose that K is a modular Borel complex on a Polish spaceX. Every balanced virtual condition in the poset PK is placid. To see this,revisit the proof of Theorem 6.4.3 and note that V [H0] ∩ V [H1] = V was theonly feature used of the mutually generic extensions V [H0], V [H1].

Corollary 9.3.6. Let X be a Borel vector space over a countable field Φ.

1. Let P be the poset of countable subsets of X linearly independent overΦ, ordered by reverse inclusion–Example 6.4.9. In the P -extension of thesymmetric Solovay model, the turbulent divide is preserved;

2. it is consistent relatively to an inaccessible cardinal that ZF+DC holds, Xhas a basis, and yet the turbulent divide is preserved.

Corollary 9.3.7. Let Γ be a Borel graph on a Polish space X.

1. Let P be the poset of acyclic countable subsets of Γ, ordered by reverseinclusion–Example 6.4.10. In the P -extension of the symmetric Solovaymodel, the turbulent divide is preserved;

2. it is consistent relatively to an inaccessible cardinal that ZF+DC holds,Γ contains a maximal acyclic subgraph, and yet the turbulent divide ispreserved.

Example 9.3.8. Suppose that E,F are Borel virtually placid equivalence re-lations on the respective Polish spaces X,Y . Suppose that λ(E) ≤ λ(F ). Thenthe E,F -collapse poset of Definition 6.6.2 is placid, and every balanced condi-tion is placid.

Example 9.3.9. Suppose that E ⊂ F are Borel equivalence relation on a Polishspaces X, with F placid. The E,F -transversal poset of Definition 6.6.5 is placid,and every balanced condition is placid.

Proof. The proof of Theorem 6.6.6 uses only one consequence of mutual generic-ity: the two mutually generic extensions V [H0], V [H1] do not share any F -classwhich is not represented in the ground model. In the case of a placid equivalencerelation E, this is implied by the assumption that V [H0] ∩ V [H1] = V by thedefinition of placidity.

9.3. THE TURBULENT DIVIDE 215

Corollary 9.3.10. Suppose that E is a Borel placid equivalence relation on aPolish space X..

1. In the extension of the symmetric Solovay model by the transversal posetfor E of Example 6.6.8, the turbulent divide is preserved;

2. it is consistent relatively to an inaccessible cardinal that ZF+DC holds, Ehas a transversal, and yet the turbulent divide is preserved.

The posets which select a structure on each E-class for a countable Borel equiv-alence relation E are placid by Example 9.3.9, resulting of many corollaries ofthe following kind.

Corollary 9.3.11. Suppose that E is a countable Borel equivalence relation ona Polish space X with infinite classes.

1. In the extension of the symmetric Solovay model by the Z-action poset ofExample 6.6.10, the turbulent divide is preserved;

2. it is consistent relative to an inaccessible cardinal that ZF+DC holds, E isan orbit equivalence of a (discontinuous) action of Z, and yet the turbulentdivide is preserved.

Example 9.3.12. Suppose that F is a Fraisse class of structures in finite rela-tional language, with strong amalgamation. Let E be a virtually placid equiva-lence relation on a Polish space X. Let P be the poset for adding an F-structureon the quotient space X with countable approximations of Definition 8.6.3.Then P is placid, and every balanced condition is placid.

Corollary 9.3.13. Let E be a Borel virtually placid equivalence relation on aPolish space X.

1. In the extension of the symmetric Solovay model by the linearization posetfor E of Example 8.6.5, the turbulent divide is preserved;

2. it is consistent relatively to an inaccessible cardinal that ZF+DC holds,the E-quotient space is linearly ordered and yet the turbulent divide ispreserved.

Example 9.3.14. Suppose that X is a Polish space and P is the Kurepa poseton it as isolated in Definition 8.4.2. Then P is placid, and every balancedcondition is placid. To see this observe that the proof of Theorem 8.4.3 usedonly the consequence V [H0] ∩ V [H1] = V for mutually generic filters H0, H1.


1. In the extension of the symmetric Solovay model by the Kurepa poset, theturbulent divide is preserved;


2. it is consistent relatively to an inaccessible cardinal that ZF+DC holds,there is a Kurepa family on X, and yet the turbulent divide is preserved.

Example 9.3.16. Let X be an uncountable Polish space and P be the associ-ated acyclic decomposition forcing of Definition ??. Then, under CH, P is placidand every balanced condition is placid. To see this observe that the proof ofTheorem ?? used only the consequence V [H0]∩V [H1] = V for mutually genericfilters H0, H1.

Corollary 9.3.17. Let X be an uncountable Polish space.

1. In the extension of the symmetric Solovay model by the acyclic decompo-sition poset, the turbulent divide is preserved;

2. it is consistent relatively to an inaccessible cardinal that ZF+DC holds,there is an acyclic decomposition of [X]2, and yet the turbulent divide ispreserved.

We conclude this section with several non-examples.

Example 9.3.18. Let E be the equivalence relation on 2ω connecting setsx, y if Σ{ 1

n+1 : n ∈ x∆y} is finite. E is a Borel orbit equivalence relation of aturbulent Polish group action; it is well-known to be pinned. Let P be the posetof countable partial E-transversals ordered by reverse inclusion. The poset Pis balanced by Corollary 6.6.4, but not placid: in fact, it is designed to collapsethe turbulent divide as it forces |E| ≤ |2ω|.

Example 9.3.19. Let X be an uncountable Polish field and F ⊂ X be a count-able subfield. The poset P of countable subsets of X which are algebraicallyindependent over F , with the reverse inclusion ordering is balanced. It is notplacid by Theorem 12.3.1 below. We do not know if in the P -extension of thesymmetric Solovay model, the turbulent divide is preserved.

Example 9.3.20. The poset P of infinite subsets of ω ordered by inclusion isbalanced. It is not placid by Theorem 12.2.3 below. We do not know if in theP -extension of the symmetric Solovay model, the turbulent divide is preserved.

9.4 The orbit divide

It is well-known that E1 is not Borel reducible to any orbit equivalence relation.It is then tempting to think that in many models which we study, |E1| cannotbe smaller than |E| where E is an orbit equivalence relation. This question iswith some success addressed in the present section. The following definitionconnects coherent sequences of generic extensions with Suslin forcings.

Definition 9.4.1. Let P be a Suslin forcing. A nest below a condition p ∈ Pis a choice-coherent sequence 〈Mn : n ∈ ω〉 of generic extensions of V and asequence 〈pn : n ≤ ω〉 so that

9.4. THE ORBIT DIVIDE 217

1. 2ω ∩Mn 6= 2ω ∩Mn+1 for all n ∈ ω;

2. p0 ≤ p1 ≤ · · · ≤ pω ≤ p where pn for each n ∈ ω is a balanced virtualcondition in Mn and pω is a balanced virtual condition in the intersectionmodel Mω =

⋂nMn.

Definition 9.4.2. Let P be a Suslin forcing. The poset P is nested balancedbelow κ if it has a nest below every condition.

As a simple initial example, let P be the poset of all countable functions from2ω to 2, ordered by reverse inclusion. The balanced conditions are classified bytotal functions from 2ω to 2. Now, let 〈Mn : n ∈ ω〉 be a coherent sequence ofmodels and p ∈Mω =

⋂nMn be a condition in P . Then the functions pn ∈Mn

for n ≤ ω obtained in the model Mn from p by extending it by zero values atevery possible point x ∈ 2ω correspond to balanced virtual conditions in everymodel Mn and p ≥ pω ≥ · · · ≥ p1 ≥ p0.To state the results of this section succintly, we establish a standard parlance.

Definition 9.4.3. (ZF) The phrase “orbit divide is preserved” denotes thefollowing statement. Let E be an orbit equivalence relation on a Polish spaceinduced as an orbit equivalence relation of a Polish group action. Then |E1| 6≤|E|.

The central theorem can now be stated easily using Convention 1.7.16.

Theorem 9.4.4. In nested balanced extensions of the symmetric Solovay model,the orbit divide is preserved.

Proof. Suppose that Y is a Polish space, Γ is a Polish group and Γ continuouslyacts on Y , inducing an orbit equivalence relation E. Suppose that κ is aninaccessible cardinal and P is a Suslin poset which is nested balanced below κ.Suppose also that W is a symmetric Solovay model derived from κ and in W ,τ is a P -name and p ∈ P is a condition forcing τ to be a function from theE1-quotient space to the E-quotient space. We must find a stronger conditionand two distinct E1 classes which are forced by the stronger condition to bemapped to the same E-class.

The objects Γ, Y , p and τ are all definable in the model W from parametersin the ground model plus a parameter z ∈ 2ω. Use the assumptions to find anintermediate forcing extension V [K] by a poset of size < κ such that z ∈ V [K]and V [K] has a nest below p. Still working in W , find the nest. This is asequence 〈Mn : n ∈ ω〉 of generic extensions of V [K] together with a sequence〈pn : n ≤ ω〉 of balanced virtual conditions below p satisfying the demands ofDefinition 9.4.1.

Use the coherence of the sequence 〈Mn : n ∈ ω〉 to find a well-ordering ≺ of2ω in M0 such that for each n ∈ ω, ≺� Mn ∈ Mn holds. Let zn ∈ 2ω be the≺-least element of Mn \Mn+1 for every n ∈ ω and let xn ∈ X = (2ω)ω be thepoint such that xn(m) is the zero binary sequence if m < n and xn(m) = zm ifm ≥ n; thus xn ∈Mn. The points xn ∈ X for n ∈ ω are pairwise E1-equivalent;


��0

��1

𝑝𝜔

��2

��3

V0

V1

V2

V3

Vω

Figure 9.1: A nest of models and balanced virtual conditions.


at the same time, they have no E1-equivalent in the model Mω =⋂nMn, since

none of the points zn belong to Mω.By the forcing theorem, for each number n ∈ ω, in the model Mn there must

be a poset Rn of size < κ, an Rn-name σn for a condition in P stronger than pnand an Rn-name ηn for an element of Y such that Rn Coll(ω,< κ) σn Pτ([xn]E1

) = [ηn]E .

Claim 9.4.5. For n ∈ ω, the pair 〈Rn, ηn〉 is an E-pin.

Proof. If this failed for some number n ∈ ω, then in the model W there wouldbe filters H0, H1 ⊂ Rn mutually generic over the model Mn such that thepoints y0 = σn/H0, y1 = σn/H1 ∈ Y are E-unrelated. Let p0 = σn/H0 andp1 = σn/H1 be conditions in the poset P ; they are compatible by the balanceof pn. By the forcing theorem applied in the models Mn[H0] and Mn[H1], theirlower bound forces in P that τ([xn]E1

) is equal simultaneously to [y0]E and[y1]E . This is impossible.

Claim 9.4.6. For n ∈ m ∈ ω, the E-pins 〈Rn, ηn〉 and 〈Rm, ηm〉 are equivalent.

Proof. If this failed for some numbers n ∈ m ∈ ω, then in the model W therewould be filters H0 ⊂ Rn, H1 ⊂ Rm mutually generic over the model Mn suchthat the points y0 = σn/H0, y1 = σm/H1 ∈ Y are E-unrelated. Let p0 = σn/H0

and p1 = σm/H1 be conditions in the poset P ; they are compatible by thebalance of pm since p0 ≤ pn ≤ pm by demand (2) of Definition 9.4.1. By theforcing theorem applied in the models Mn[H0] and Mm[H1], their lower boundforces in P that τ([xn]E1

) = [y0]E and τ([xm]E1) = [y1]E . This is impossible

since xn E1 xm holds.

By Theorem 4.3.6, there is an E-pin 〈R, η〉 ∈ Mω which is equivalent to allthe pins 〈Rn, ηn〉 for n ∈ ω. Still in Mω, find a poset Q generating the modelM0, a Q-name x for the sequence x0, and a Q-name R0 for the poset R0 . By theforcing theorem, there must be a condition q ∈ Q which forces the following: xhas no E1-equivalent in Mω, R0 σ0 ≤ pω and R× R0 η E η0. Note that thelast statement means that the name η0 is E-pinned on the iteration Q � q ∗ R0.In the model W , let H0, H1 ⊂ Q � q ∗ R0 be filters mutually generic over themodel Mω, and write p0 = σ0/H0, p1 = σ0/H1, x0 = x/H0, x1 = x/H1,y0 = η0/H0 and y1 = η0/H1. The balance of the condition pω implies thatp0, p1 ∈ P are compatible conditions. Since E1 is a pinned equivalence relation,the points x0, x1 ∈ X are E1-unrelated. Since the name η0 was pinned on theiteration Q � q∗R0, the points y0, y1 are E-related. In total, in the model W , thelower bound of the conditions p0, p1 forces that τ([x0]E1

) = τ([x1]E1) = [y0]E ,

completing the proof.

It turns out that one large class of nested balanced posets has already beenisolated in Definition 9.2.1:

Example 9.4.7. Every compactly balanced Suslin poset is nested balanced.


Proof. Let P be a compactly balanced Suslin poset and p ∈ P be a condition.Let U ⊂ P(ω) modulo finite and 〈xn : n ∈ ω〉 be mutually generic ultrafilter andSacks reals added with full support product. Let 〈Mn : n ∈ ω〉 be the sequenceof models determined by Mn = V [U, ym : m ≥ n]; this is a choice-coherentsequence by Theorem 4.3.5 or Example 4.3.4 applied in the model V [U ]. Wewill produce a descending sequence of balanced virtual conditions as requiredby Definition 9.4.1.

It is well known [84] that the Sacks real product adds no independent re-als, and therefore, by a genericity argument, U generates an ultrafilter on ωin the model M0. The generating set of U is present already in the modelMω =

⋂nMn. For each number n ∈ ω, use the extension property (2) of Def-

inition 9.2.1 repeatedly to find a sequence 〈p(ω, n), p(m,n) : m ≤ n〉 such thatp(ω, n) is a balanced virtual condition in Mω, each p(m,n) is a balanced vir-tual condition in the model Mm, and p ≥ p(ω, n) ≥ p(n, n) ≥ p(n − 1, n) ≥p(n − 2, n) ≥ . . . ; also, demand that each virtual condition p(m,n) is selected≺-least where ≺ in M0 is some coherent well-ordering of the set of virtual condi-tions on P . It is not difficult to see that the resulting system is coherent in thesense that for each n ≤ ω, the system 〈p(m, k) : m = ω and k ∈ ω or k ≥ m ≥ n〉belongs to the model Mn.

Finally, use the compactness of the space of balanced virtual conditions todefine pn for n ≤ ω to be the U -limit of the sequence 〈p(n, k) : k ≥ n〉. Eachpn is a balanced virtual condition in the model Mn, m ≤ n ≤ ω implies thatpm ≤ pn ≤ p by the closedness demands (1) and (3) of Definition 9.2.1. Itfollows that the sequence 〈pn : n ≤ ω〉 is as required in Definition 9.4.1.

Corollary 9.4.8.

1. Let P be the poset P(ω) modulo finite. In the P -extension of the symmetricSolovay model, the orbit divide is preserved.

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds, thereis a Ramsey ultrafilter on ω, and the orbit divide is preserved.

Proof. This is a conjunction of Example 9.2.4 and Theorem 9.4.4.

Question 9.4.9. Is there a Borel (or even Fσ) ideal I on ω such that theexistence of an ultrafilter disjoint from I implies in ZF+DC that |E1| ≤ |E| forsome orbit equivalence relation E?


1. In the extension of the Solovay model by the E-linearization poset of Ex-ample 8.6.5, the orbit divide is preserved.

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds, theE-quotient space is linearly ordered, and the orbit divide is preserved.

Proof. This is a conjunction of Example 8.6.5, Example 9.2.11 and Theorem 9.4.4.


Corollary 9.4.11. Let Γ,∆ be abelian Polish groups, with Γ uncountable andtorsion-free and ∆ compact, nontrivial, and divisible.

1. Let P (Γ,∆) be the poset of Definition 8.2.1. In the P (Γ,∆)-extension ofthe symmetric Solovay model, the orbit divide is preserved.

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds,there is a discontinuous homomorphism from Γ to ∆, the orbit divideis preserved.


Question 9.4.12. Does the existence of a discontinuous homomorphism from Rto R imply in ZF+DC that |E1| ≤ |E| holds for some orbit equivalence relationE, or for E = F2 in particular? That E1 has a countable complete section?


1. In the extension of the Solovay model by the Kurepa poset of Defini-tion 8.4.2, the orbit divide is preserved.

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds, thereis a Kurepa family on X, and the orbit divide is preserved.


Another group of nested balanced posets is obtained from orbit equivalencerelations:

Example 9.4.14. Let E ⊂ F be Borel equivalence relations on a Polish spaceX such that F is pinned and Borel reducible to an orbit equivalence relation. LetP be the E,F -transversal poset of Definition 6.6.5. Then P is nested balanced.

Proof. By Theorem 6.6.6, the balanced virtual conditions are classified by func-tions selecting, for each F -class, a single virtual E-class which is forced to be asubset of the F -class. Let p ∈ P be a condition. To find a nest below p, considerany choice-coherent sequence 〈Mn : n ∈ ω〉 consisting of generic extensions ofthe ground model such that for every n ∈ ω, 2ω ∩Mn \Mn+1 6= 0–the choice-coherent sequence of models obtained from the infinite product of Sacks realswill do. Let ≺ be a coherent well-ordering of the space X. To find a coherentsequence of balanced virtual conditions, in the model M0 consider the functionf on the F -quotient space defined as follows: if c is an F -class mentioned inp, let f(c) be the E-class d ⊂ c which p selects. If c is a class represented inMω but not mentioned in p, then let f(c) be the E-class of the ≺-first repre-sentative of c in the model Mω. If n ∈ ω is the largest number such that c isrepresented in Mn, then let f(c) be the E-class of the ≺-first representative ofc in the model Mn. Now, by Theorem 4.3.6, this defines the function f on allF -equivalence classes represented in M0. It is also clear that for each n ≤ ω,f � Mn ∈ Mn. For each n ≤ ω, let pn be the virtual balanced condition in themodel Mn associated with the function f � Mn by Theorem 6.6.6. The system〈Mn : n ∈ ω, pn : n ≤ ω〉 is the sought nest below p.


Corollary 9.4.15. Let E be a Borel pinned orbit equivalence relation on aPolish space X.

1. Let P be the transversal poset of Example 6.6.8. In the P -extension of thesymmetric Solovay model, the orbit divide is preserved.

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds, Ehas a transversal, and the orbit divide is preserved.


The posets selecting a structure on each E-class for a countable Borel equiva-lence relation E are always nested balanced by virtue of Example 9.4.14. Thefollowing is a sample corollary.

Corollary 9.4.16. Let E be a countable Borel equivalence relation on a Polishspace X, with all orbits infinite.

1. Let P be the Z-action poset of Example 6.6.10. In the P -extension of thesymmetric Solovay model, the orbit divide is preserved.

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds, E isinduced as an orbit equivalence relation of a discontinuous Z-action, andthe orbit divide is preserved.

The determination of the nested balanced status of the posets adding amaximal K-set for a Borel simplicial complex K is much more challenging. Theultimate limitation is the surprising Corollary 9.4.30 below, showing in ZF+DCthat if R has a Hamel basis, then E1 has a complete countable section, andtherefore |E1| ≤ |F2| holds. Thus, we have to resort to partial results.

Example 9.4.17. Let Γ be a Borel graph on a Polish space X with Borelcoloring number ℵ1. Let P = PΓ be the poset of countable Γ-anticliques orderedby reverse inclusion. Then P is nested balanced.

Proof. Let p ∈ P be a condition; we must produce a nest below p ∈ P . Let〈Mn : n ∈ ω〉 be a choice-coherent sequence of generic extensions such that M0

is an extension of the ground model by proper forcing, and for all n ∈ ω the set2ω ∩Mn \Mn+1 is nonempty. We will show that there is a sequence 〈pn : n ∈ ω〉of balanced conditions satisfying the demands of Definition 9.4.1.

Theorem 6.3.3 classified the balanced conditions in the poset P in terms ofweakly maximal Γ-anticliques. Thus, it will be enough to produce a set A ⊂ Xin the model M0 which contains p as a subset and such that for each n ≤ ω,the set A ∩Mn belongs to Mn and is a weakly maximal Γ-anticlique there. Tocut the notational clutter, we ignore the condition p ∈ P . Let ~Γ be a Borelorientation of Γ such that the ~Γ-outflow o(y) of every vertex y ∈ X is countable.Consider the set B ⊂ X of all points x ∈ X for which there is a countable seta ⊂ X such that


• either for every countable set b ⊂ X there is y /∈ b such that x ∈ o(y) ⊂ a;

• or for every countable set b ⊂ X there is y ∈ X such that 〈y, x〉 ∈ ~Γ ando(y) \ a is a nonempty set disjoint from b.

A careful complexity computation shows that the set B is Σ12. The following

claim uses the properness assumption:

Claim 9.4.18. Whenever n ∈ ω is a number and y ∈ X ∩Mn \Mn+1 is a point

and 〈y, x〉 ∈ ~Γ, then x ∈ B or x ∈Mn \Mn+1.

Proof. Suppose that x ∈Mn+1; we must show that x ∈ B. The set o(y)∩Mn+1

is countable in Mn; by the properness assumption, it is covered by a set a ∈Mn+1 which is countable in Mn+1. We claim that a is a witness to x ∈ B.Indeed, if o(y) ⊂ Mn+1 then x ∈ B by the first item in the definition of B–thepoint y is not in Mn+1 and so it does not belong to any countable set in Mn+1.On the other hand, if o(y) ⊂ Mn+1 then x ∈ B by the second item in thedefinition of B–the nonempty set o(y) \ a is disjoint from any countable set inMn+1.

Now, work in M0. Let ≺ be a coherent well-ordering of P(X), and for everynumber n ∈ ω let Cn ∈ Mn be the ≺-least maximal Γ-anticlique in the set(X \ B) ∩ Mn \ Mn+1. Let also Cω ∈ Mω be the ≺-maximal Γ-anticliquein (X ∩ Mω) \ B extending p. Let A =

⋃n≤ω Cn; we claim that for each

n ≤ ω, the set A ∩Mn belongs to Mn and is weakly maximal there. Note thatA∩Mn ∈Mn follows immediately from the coherence of the choices above. Theclaim implies that A∩Mn is a Γ-anticlique and so a maximal Γ-anticlique in theset (X \B)∩Mn. It follows directly from the definition of a weakly maximal set(Definition 6.3.2) that A ∩Mn is a weakly maximal set in Mn as desired.

Corollary 9.4.19. Let Γ be a Borel graph on a Polish space X of Borel coloringnumber ℵ1.

1. In the PΓ extension of the symmetric Solovay model and the orbit divideis preserved;

2. it is consistent relative to an inaccessible cardinal that ZF+DC holds, thereis a maximal Γ-anticlique and the orbit divide is preserved.

Example 9.4.20. Let X be a Polish space and let G be a Borel graph on X.Let K be the Borel simplicial complex on G of finite acyclic subsets of G, andlet P = PK. Then P is nested balanced.

Proof. Let Qm for each m ∈ ω be the countable support product of ℵ1 manySacks posets, and for each n ∈ ω let Rn be the countable support productΠm≥nQm, with the natural projection maps. Let H0 ⊂ R0 be a generic filter,and for each n ∈ ω let Hn = H ∩Rn. By Example 4.3.4 and Theorem 4.3.5, thesequence 〈V [Hn] : n ∈ ω〉 is a choice-coherent sequence of models of ZFC; writeMn = V [Hn] and Mω =

⋂nMn. In view of Theorem ??, to complete the proof


of the example it will be enough for an arbitrary condition p ∈ P ∩ V to findsets Am ⊂ G for m ≤ ω forming an inclusion-descending sequence such thatp ⊂ Aω and Mm |= Am ⊂ G is a maximal acyclic subset of G for every m ≤ ω.

Write Xω = X∩Mω, and for each m < n write Xmn = Xω∪(X∩Mm \Mn).The following elusive claim is the main reason for the choice of the posets Qm.

Claim 9.4.21. Let l < m < n be natural numbers and x0, x1 ∈ Xmn be vertices.If x0, x1 are connected by a path in the graph G � Xln, then they are connectedby a path in the graph G � Xmn.

Proof. It will be enough to show that if x0, x1 are connected by a path in thegraph G � Xln whose vertices except for x0, x1 all belong to Xlm, then they areconnected by a path in the graph G � Xmn. To this end, work in Mm and letA = {a ∈ [X]<ℵ0 : there is a G-path connecting x0, x1 using only the verticesin a}.

The set A ⊂ [X]<ℵ0 is Borel. It cannot be punctured by a countable set by aMostowski absoluteness argument: in the model Ml there is a G-path betweenx0, x1 using no vertices in Mm and therefore no vertices in any given countableset in Mm. By the work on puncture sets [16, Theorem 21] applied in the modelMm, there is a perfect set B ⊂ A in the model Mm consisting of pairwise disjointsets. By a standard fusion argument with the product of Sacks forcing, there isa countable set c of Sacks reals added by the filter Hm such that B has a codein the model V [c] ⊂Mm. Since the poset Qm is a product of uncountably manycopies of Sacks forcing, in Mm there has to be an element a ∈ B which doesnot belong to the model Mn[c] ⊂Mm. Since the sets in B are pairwise disjoint,every element of a would reconstruct a over the model Mn[c]. Therefore, noelement of a belongs to the model Mn[c]; in particular, a∩Mn = 0 and a yieldsthe desired G-path between x0 and x1 using only vertices in Xmn.

Let ≺ be a coherent well-ordering of all subsets of G in M0. Let Aω ⊂ Gbe a maximal acyclic subset of G in the model Mω such that p ⊂ Aω. Foreach m ∈ ω, let Amm+1 ∈ Mm be the ≺-first set in the model Mm which is amaximal acyclic subset of G � Xmm+1 and extends Aω.

Now, by induction on n−m, for m < n define Amn ∈Mm to be the ≺-firstset in the model Mm which is a maximal acyclic subset of G � Xmn and extendsboth Am+1,n and Am,n−1. To see that this is possible, simultaneously argue byinduction on n−m that if m ≤ m′ < n′ ≤ n then Am′n′ ⊂ Amn and moreover,the set Am+1,n∪Am,n−1 does not contain a cycle. The former statement is clear.The latter statement requires the claim. A putative cycle c ⊂ Am+1,n ∪Am,n−1

has to contain some edges from both sets by the acyclicity induction hypothesis.Choose an inclusion-maximal contiguous part d ⊂ c of the cycle consisting ofedges in Am,n−1 \ Am+1,n. The end-nodes of d, denote them by x0, x1, mustbe distinct because the cycle must use some edges from the set Am+1,n as well.It must also be the case that x0, x1 ∈ Xm,n−1 ∩ Xm+1,n = Xm+1,n−1. Thepath d connects the nodes x0, x1 in the graph G � Xm,n−1; by the claim then,they have to be connected by a path in the graph G � Xm+1,n−1 as well, and


therefore by a path e in the set Am+1,n−1. Then e ∪ d forms a cycle in the setAm,n−1, violating the induction hypothesis.

In the end, let Am =⋃n>mAmn. It is clear from the coherence of the

well-ordering ≺ that Am ∈ Mm. The construction also guarantees that Mm |=Am ⊂ G is a maximal acyclic subset, and m < n implies An ⊂ Am. The proofis complete.

Corollary 9.4.22. Let X be a Polish space and let G be a Borel graph on X.

1. Let P be the poset of Example 6.4.10 for adding a maximal acyclic subsetof G. In the P -extension of the symmetric Solovay model, the orbit divideis preserved;

2. it is consistent relative to an inaccessible cardinal that ZF+DC holds, Ghas a maximal acyclic subset, and the orbit divide is preserved.

Example 9.4.23. Let Γ be a Borel graph on a Polish space X of countablecoloring number. The coloring poset P = PΓ of Definition 8.1.1 is nested bal-anced.

Proof. We identify P with its dense Suslin subset identified in Theorem 8.1.2.Let p ∈ P be a condition, and let 〈Mn : n ∈ ω〉 be a nontrivial choice coherentsequence of generic extensions of V , with Mω =

⋂nMn. We will produce a

decreasing sequence 〈p : n ≤ ω〉 of balanced virtual conditions below p for therespective models Mn. Let 〈an : n ≤ ω〉 be a recursive sequence of pairwisedisjoint infinite subsets of ω.

Claim 9.4.24. Fix n ≤ ω. In the model Mn, there is a total Γ coloring d : X →ω which extends p and at all points x ∈ X \ dom(p), d(x) ∈ an.

Proof. Work in Mn. Fix a coloring e : X → ω. Let an =⋃i bi be a partition

of an into infinitely many sets of size m. Let d : X → ω be a function suchthat p ⊂ d and for all x /∈ X \ dom(p), d(x) is a number in e(x) which is notone of the (fewer than m many) colors {p(y) : 〈x, y〉 ∈ Γ}. This is the desiredcoloring.

Let ≺ be a coherent wellordering of P(X × ω), and for each n ≤ ω let dn bethe ≺-least coloring as in the claim in the model Mn. Finally, in the model M0,define a map c : X → ω as follows: c(x) = dn(x) if x /∈ dom(p) and n ≤ ω isthe largest number such that x ∈Mn, and c(x) = p(x) if x ∈ dom(p). It is notdifficult to check that c is a Γ-coloring extending p. Moreover, the coherence ofthe well-ordering ≺ guarantees that for each n ≤ ω, c �Mn ∈Mn holds. In eachmodel Mn for n ≤ ω, the coloring c � Mn is associated with a balanced virtualcondition pn for the poset PΓ by Theorem 8.1.2. The sequence 〈pn : n ≤ ω〉 isas desired.

Corollary 9.4.25. Let Γ be the graph on R2 connecting points x0, x1 if theyhave a nonzero rational distance.


1. In the PΓ-extension of the symmetric Solovay model, the orbit divide ispreserved;

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds, thechromatic number of Γ is countable, and the orbit divide is preserved.

Example 9.4.26. Let X be a Borel vector space over a countable field Φ. LetP be the 3-Hamel decomposition forcing of Definition 8.1.11. The poset P isnested balanced.

Proof. Let J be the ideal on ω from which the poset P is constructed. Let p ∈ Pbe any condition. Let 〈Mn : n ≤ ω〉 be a generic choice-coherent sequence suchthat for every n ∈ ω, the set 2ω ∩Mn \Mn+1 is nonempty. Let ≺ be a coherentwell-ordering of all functions from X to ω. Let {an : n ≤ ω} ∈ V be a collectionof pairwise disjoint infinite sets such that

⋃n an ∈ J . Use Claim 8.1.14 in each

model Mn to show that in Mn, there is a 3-Hamel decomposition extending psuch that all points not in dom(p) get a color in the set an; let dn be the ≺-leastsuch a coloring. Now, working in Mn, let cn : X → ω be the map defined bycn(x) = dω(x) if x ∈Mω, and cn(x) = dm(x) if x ∈Mm \Mm+1.

We claim that cn is a 3-Hamel decomposition. To see this, suppose thatx0, x1, x2 are distinct nonzero points points in a nontrivial linear combinationresulting in 0. Let m ≤ ω be the largest index such that one of the points,say x0, is in Mm. Either all three points are in Mm but not in Mm+1 andthen the triple is not monochromatic as dm is a Γ-coloring. Or, the other twopoints are in Mk \Mk+1 for some k < m. In such a case the triple is again notmonochromatic: either x0 ∈ dom(p) and then the failure of monochromaticityfollows from the assumption that dk is a 3-Hamel decomposition, or x0 /∈ dom(p)and then cn(x0) ∈ am and cn(x1) ∈ ak and the sets am, ak are disjoint.

Now, it is not difficult to see that 〈cn : n ∈ ω〉 is a coherent sequence of3-Hamel decompositions. A 3-Hamel decomposition is a balanced condition forP by Theorem ??. Finally, p ≥ cω ≥ · · · ≥ c2 ≥ c1 ≥ c0 holds in the poset Pas the sets an for n ≤ ω all belong to the ideal J . Thus, 〈Mn, cn : n ≤ ω〉 is therequired nest below the condition p.

Corollary 9.4.27. Let X be a Borel vector space over a countable field Φ. LetP be the 3-Hamel decomposition forcing of Definition 8.1.11.

1. Let P be the 3-Hamel decomposition forcing. In the P -extension of thesymmetric Solovay model, the orbit divide is preserved;

2. it is consistent relative to an inaccessible cardinal that ZF+DC holds, thereis 3-Hamel decomposition of X, and the orbit divide is preserved.

In view of Example 8.1.15 it is consistent relative to an inaccessible cardinalthat ZF+DC holds, the orbit divide is preserved, and there is a decompositionof R2 into countably many pieces neither of which contains the vertices of anequilateral triangle.


Now it is time to show that the existence of certain combinatorial objectsimplies that the orbit divide breaks. In the following theorem and corollaries,we neglect the real parameter necessary to define the Polish spaces in question.For the rest of the section, put X = (2ω)ω.

Theorem 9.4.28. (ZF) Suppose that there is a Polish space Y and a set A ⊂ Ysuch that the preordering ≤ on X, defined by x1 ≤ x0 if x1 ∈ L[x0, A], is well-founded. Then there is a countable complete section for E1.

Proof. For each point x ∈ X and each number m ∈ ω, write x \ m for theelement of X defined by (x \ m)(n) = x(n) if n ≥ m, and (x \ m)(n) =theinfinite sequence of zeroes if m < n. For each number m ∈ ω write Mm(x)for the model L[x \m,A]. By the well-foundedness assumption, there must ben ∈ ω such that for all m ≥ n the models Mm(x) are the same. Note that ifx0, x1 ∈ X are E1-related points, the stable value of the models Mm(x0) andMm(x1) is the same. We would like to define the complete section D ⊂ Xby setting x ∈ D if x belongs to

⋂mMm(x). The stabilization feature shows

that D has nonempty intersection with each E1-class. However, a more precisedefinition of the complete section is necessary to make the conclusion that theintersection with each E1-class is countable.

Observe first that by a standard condensation argument the model Mm(x) isa model of ZFC+CH, and its constructibility ordering orders its elements of X

in ordertype ωMm(x)1 . For natural numbers n ≤ m ∈ ω define ρnm(x) to be the

index of x\n in the constructibility well-ordering of Mm(x) if x\n ∈Mm(x), andlet ρnm(x) = 0 otherwise. Let α(x) = lim supn supm≥n ρnm(x). The definitionof the ordinal α(x) does not depend on the choice of x within its equivalenceclass by the stabilization assumption. Also, α(x) is a countable ordinal. To seethis, work in the model M0(x) and evaluate the ordinal α(x) there. Since M0(x)

is a model of choice, its ω1 is regular and so α(x) ∈ ωM0(x)1 . Since ω

M0(x)1 ≤ ω1,

the countability of α(x) follows.Finally, let C = {x ∈ X : ∃n x = x \ n and for all m ≥ n, x \ n ∈ Mm(x)

and ρnm(x) ≤ α(x)}. To see that the set C meets every E1 class in a nonemptycountable set, let z ∈ X be arbitrary. The definition of the ordinal α(z) showsthat there is a number n ∈ ω such that for all m ≥ n, z \ n ∈ Mm(z) and isenumerated before α(z) there; clearly, letting x = z \ n we get x ∈ C ∩ [z]E1 .Also, note that C ∩ [z]E1

⊂ {x ∈ X : ∃n x ∈ Mn(z) and the index of x in theconstructibility well-ordering of Mn(z) is ≤ α(z)} and observe that the latterset is countable as the ordinal α(z) is countable.

Corollary 9.4.29. (ZF) Let Y be an uncountable Polish space. If there is anacyclic decomposition of [Y ]2 then E1 has a countable complete section.

In particular, in ZF the existence of an acyclic decomposition implies |E1| ≤ |F2|.

Proof. Let c : [Y ]2 → ω be the acyclic decomposition. Fix a countable basis forthe space Y and let A = {〈z0, z1, O,m〉 : z0, z1 ∈ Z are distinct points, O ⊂ Y isa basic open set, m ∈ ω, and there is y ∈ O such that c(y, z0) = c(y, z1) = m}.


Let ≤ be the preordering on X defined by x1 ≤ x0 if x1 ∈ L[x0, A]. In view ofTheorem 9.4.28 it will be enough to show that ≤ is well-founded.

To do this, consider the map π : X → ω1 defined by π(x) = ωL[x,A]1 . It

will be enough to show that x1 < x0 implies π(x1) ∈ π(x0). Suppose towardscontradiction that this fails for some x1 < x0, and write M1 = L[x1, A] andM0 = L[x0, A]. Since M1 is a class of M0, it must be the case that ωM1

1 ≤ ωM01 ;

by the contradictory assumption, the equality in fact prevails. Thus, in themodel M0 the set Y ∩ M1 is an uncountable proper subset of Y ∩ M0. Lety ∈ Y ∩ (M1 \M0) be an arbitrary point.

By a counting argument in the model M0 there must be a number m ∈ ωand distinct points z0, z1 ∈ Y ∩M1 such that c(y, z0) = c(y, z1) = m. If y ∈ Ywas the unique point satisfying these equalities then it can be constructed fromA and the points z0, z1; as such, it would belong to the model M1, which is notthe case. Thus, there is a point y′ ∈ Y distinct from y such that c(y′, z0) =c(y′, z1) = m. However, then the 4-cycle consisting of edges connecting thevertices z0, y, z1, y

′, z0 in this order is monochromatic, a contradiction.

Corollary 9.4.30. (ZF) Let Y be an uncountable Polish vector space over acountable field Φ. If there is a basis for Y over Φ then there is a countablecomplete section for E1.

In particular, in ZF the existence of a Hamel basis for R over Q implies |E1| ≤|F2|.

Proof. Let d be an invariant complete metric on Y . Let B ⊂ Y be a basis forY over F . Let A = {〈y,Oi : i ∈ k〉 : y ∈ Y,Oi ⊂ Y are basic open sets andthere are points yi ∈ Oi ∩ b such that y is a linear combination of yi : i ∈ kwith nonzero coefficients in Φ}. Define a preordering on the space X by settingx1 ≤ x0 if x1 ∈ L[x0, A,B]. It will be enough to show that the preordering ≤ iswell-founded. A reference to Theorem 9.4.28 then concludes the proof.

Suppose towards contradiction that the ordering ≤ is ill-founded. The firstorder of business is to produce a sequence 〈xn : n ∈ ω〉 such that for each m ∈ ω,〈xn : n ≥ m〉 ∈ L[xm, A,B] and xm /∈ L[xm+1, A,B]. To do this, let x0 ∈ X beany element such that there is a strictly decreasing infinite sequence in ≤ belowx0, and by recursion define xn+1 ∈ X to be the first (in the constructibilityordering of L[xn, A,B]) point in L[xn, A,B] such that xn /∈ L[xn+1, A,B] andthere is a strictly decreasing sequence in ≤ below xn+1. The point here is thatthe ordering is defined in an absolute way, and its well-foundedness is absoluteinto the models concerned.

Write Mn for the model L[xn, A,B] and Mω for⋂nMn. As an initial ob-

servation, note that the set B ∩Mn is a basis in Mn: for every y ∈ Y ∩Mn, theunique linear combination of elements of B with nonzero coefficients yielding ycan be constructed from y and C, so belongs to Mn. Now, by recursion on i ∈ ωbuild points yi ∈ Y and numbers ni ∈ ω so that n0 = 0 and for every i ∈ ω,yi ∈ Mni is the first point in the constructibility well-ordering of Mni which iswithin d-distance 2−i of the zero element of Y and does not belong to Mni+1.Such a point must exist since the 2−i-neighborhood of the zero element is an

9.5. THE EKσ DIVIDE 229

uncountable open subset of the Polish space Y , and Mni+1 does not contain allreals of Mni . The point yi is expressed as a unique linear combination φi ofelements of B ∩Mni . Finally, let ni+1 ∈ ω be the first number n greater thanni such the linear combination φi uses no elements of the set Mn \Mω.

Note that for each i ∈ ω the sequence 〈yj , nj : j ≥ i〉 belongs to the modelMni . Let zi = Σj≥iyj = limk Σk≥j≥iyk. The limit exists as the metric d isinvariant and complete and the points yi converge to zero fast. It is also clearthat zi ∈ Mni , and zi − zi+1 = yi. Since zi+1 ∈ Mni+1

and yi /∈ Mni+1both

hold, we conclude that zi /∈Mni+1holds.

Now, the point z0 ∈ Y can be expressed as a linear combination ψ of elementsof B ∩M0. Let i ∈ ω be a number so large that the combination ψ uses noelements of Mni\Mω. Observe that zi = z0−Σj<iyj and the linear combinationsφj yielding the points yj for j ∈ i use no elements of Mni \Mω. In other words,the point zk ∈Mk can be expressed as a linear combination of elements of B\M0

which uses no elements of Mni \Mω. This combination (after cancellations) isunique as B is a basis, and it uses only elements of Mni since B∩Mni is a basisin Mni . Thus, it uses only elements of Mω, so zi ∈Mω. This ontradicts the lastsentence of the previous paragraph.

Note that the Polish assumption on the vector space Y is necessary in Corol-lary 9.4.30. One can consider the (non-Polishable) group Y of finite subsets of2ω equipped with the symmetric difference operation as a vector space over thebinary field. This vector space has a basis in ZF, namely the set of all singletons.Thus, the existence of the basis for this particular vector space does not implyin ZF the existence of a complete countable section for E1.

9.5 The EKσdivide

Recall that EKσ is the equivalence relation on the space of all functions inωω pointwise dominated by the identity function, connecting functions x0, x1

if the function x0 − x1 is bounded. The central position of this equivalencerelation is documented by the fact that it is Kσ, and that every Kσ-equivalencerelation is Borel reducible to it by a result of Rosendal [74]. It is well-knownthat the equivalence relation Eω0 is not Borel reducible to EKσ [48, Lemma6.1.1], and among the equivalence relations classifiable by countable structures,it even has a central place in this regard. In this section, we show that incertain extensions of the symmetric Solovay model, this non-reducibility resultis translated to a cardinality preservation result. The reader should note thatin ZF, the inequality |E0| ≤ |2ω| implies |Eω0 | ≤ |(2ω)ω| = |2ω|, and thereforepreserving the EKσ -divide is strictly more difficult than preserving the smoothdivide.

The preservation of the relation |Eω0 | 6≤ |EKσ | uses the following generaliza-tion of the mutual genericity motivated by Proposition 1.7.8.

Definition 9.5.1. Let V [H0], V [H1] be generic extensions in some ambientextension of V . We say that V [H0], V [H1] are mutually generic over a pod if


1. for every ordinal α, Vα ∩ V [H0] ∩ V [H1] ∈ V [H0] ∩ V [H1];

2. for any disjoint sets a0 ∈ V [H0] and a1 ∈ V [H1] which are subsets ofV [H0] ∩ V [H1], there are disjoint sets b0, b1 ∈ V [H0] ∩ V [H1] such thata0 ⊂ b0 and a1 ⊂ b1.

The pod is the intersection V [H0] ∩ V [H1].

It is not hard to see that the first item implies that the intersection V [H0] ∩V [H1] is a model of ZF, since it is weakly universal and closed under the Godelfunctions [45, Theorem 13. 9]. If the intersection is in fact a model of ZFC, thenthe extensions V [H0], V [H1] are mutually generic over it by Proposition 1.7.8.However, we will be interested exactly in the situations where the intersectionfails to satisfy even DC. The following is a simple observation which will becritical later.

Proposition 9.5.2. Let V [H0], V [H1] be extensions mutually generic over apod. Every EKσ -class represented in both V [H0], V [H1] is represented in V [H0]∩V [H1].

Proof. Let x0 ∈ V [H0] and x1 ∈ V [H1] be points in ωω below the identityfunction which are EKσ -related. Find a natural number m ∈ ω such that for allk ∈ ω, |x0(k)−x1(k)| ≤ m. Let a0 ⊂ ω×ω be the set x0, and let a1 ⊂ ω×ω bethe set {〈k, l〉 : |l − x1(k)| > m}. Thus, a0 ∈ V [H0] and a1 ∈ V [H1] are disjointsets, and therefore can be separated by some disjoint sets b0, b1 ⊂ ω × ω inV [H0]∩V [H1] by the mutual genericity assumption. The vertical sections of theset b0 are all nonempty and the function y ∈ 2ω, which for each n ∈ ω indicatesthe smallest element of the vertical section (b0)n, belongs to V [H0] ∩ V [H1].Since y ∩ a1 = 0, it must be the case that y is EKσ -related to both x0, x1 asdesired.

The conclusion of the proposition fails for such simple pinned equivalence rela-tions as Eω0 , and this is exactly the point exploited in this section. We will needa notion of pod balance.

Definition 9.5.3. A Suslin poset P is pod balanced if for every pair V [H0],V [H1] of extensions mutually generic over a pod and every pair of conditionsp0 ∈ V [H0] and p1 ∈ V [H1], if p0, p1 ∈ P are incompatible, then there areanalytic sets A0, A1 ⊂ P coded in V [H0] ∩ V [H1] such that ΣA0, ΣA1 areincompatible in P and p0,ΣA0 are compatible and p1,ΣA1 are compatible.

Theorem 9.5.4. In compactly balanced, pod balanced extensions of the sym-metric Solovay model, |Eω0 | 6≤ |EKσ | holds.

Proof. The main technical tool used in the proof is the countable support prod-uct Q of countably many copies of the Vitali forcing. We first study the productQ in its own right.

Fact 9.5.5. The poset Q is proper, bounding, and adds no independent reals.


Proof. The first two assertions are well-known, and follow for example from [?,Theorem 5.6] or [101, Theorem 5.2.6]. The third assertion is more difficult. Itcan be derived from the work of [84, Section 4], and it appears explicitly in theforthcoming [104].

The poset Q is designed to add an interesting model of ZF. In the Q-extension,let 〈xn : n ∈ ω〉 be the sequence of points Vitali-generic over the ground modeladded by the product Q. Let A ⊂ ω × 2ω be the relation consisting of allpairs 〈n, x〉 such that x is E0-related to xn. For each number n ∈ ω, let Vn bethe model of all sets hereditarily definable from xm for m ∈ n, A, and someparameters in the ground model. Also, let M be the model of all sets hereditarilydefinable from the set A and its elements, and parameters in the ground model.We have the following:

Claim 9.5.6. 1. V0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂M ;

2. the sequence 〈Vn : n ∈ ω〉 belongs to M ;

3. every set of ordinals in M belongs to⋃n Vn;

4. the model M does not contain any uniformization of the set A.

Note also that the models Vn satisfy the axiom of choice, while the model Mfails even DC, as item (3) shows.

Proof. The first item follows directly from the definitions of the models Vn andM : if m ∈ n ∈ ω are numbers then there are fewer definitions of elementsof the model Vm than of elements of the model Vn, and certainly fewer thanthe definitions of elements the model M . For (2), the sequence 〈Vn : n ∈ ω〉 isdefinable from the set A, since in the definition of the model Vn it certainly doesnot matter which E0-equivalents of the points xm for m ∈ n one uses as theparameters. For (3), any definition of a set of ordinals in the model M uses onlyfinitely many parameters, and then must belong to the model Vn for a numbern large enough so that the parameters of the definition are definable from xmfor m ∈ n, the set A, and some elements of the ground model.

Lastly, for (4), return to the ground model and assume towards a contra-diction that some condition in Q forces (4) to fail. Then, in the poset Q, theremust be a condition q, numbers n, k ∈ ω and a formula φ such that q forces thatthe formula φ uses only parameters xm for m ∈ n, A, and some parameters inthe ground model, it defines a uniformization y of the set A, and y(n) agreeswith xn from k on. Let 〈xm : m ∈ ω〉 be a Q-generic sequence of points meetingthe condition q. By a basic analysis of the Vitali forcing in [101, Section 4.7.1],there is also a point x′n ∈ 2ω E0-related to xn which differs from xn at somepoint past k and such that 〈xm : m 6= n, x′n〉 is a Q-generic sequence meeting thecondition q. Applying the forcing theorem to the two sequences 〈xm : m ∈ ω〉and 〈xm : m 6= n, x′n〉, we see that the formula φ should define a uniformiza-tion y of the set A such that y(n) is equal to both xn and x′n beyond k. Thisis impossible though as the two sequences xn, x

′n do differ at some point past

k.


Now, towards the proof of the theorem. Write E = Eω0 and F = EKσ. Letκ be an inaccessible cardinal. Let P be a Suslin forcing which is compactlybalanced and pod balanced below κ. Let W be a symmetric Solovay modelderived from κ and work in the model W . Let p ∈ P be a condition and τ be aP -name such that p forces τ to be a function from the E-quotient space to theF -quotient space. We must find a condition q ≤ p and distinct E-classes whichare forced by q to have the same τ -image.

The condition p and the name τ must be definable from a parameter z ∈ 2ω

and additional parameters from the ground model. Let V [K] be an intermediateforcing extension obtained by a poset of size < κ such that z ∈ V [K]. Working inthe model V [K], let Q0 be the poset P(ω) modulo finite, and let Q1 be the prod-uct of countably many copies of the Vitali forcing as discussed in the preamble ofthe present proof. Let 〈U, xn : n ∈ ω〉 be objects generic over V [K] for the posetQ0 ×Q1; in particular, U is an ultrafilter on ω, and 〈xn : n ∈ ω〉 is a sequenceof points in 2ω. Since the poset Q0 is σ-closed, the models V [K] and V [K][U ]compute the poset Q1 in the same way, in particular V [K][U ][xn : n ∈ ω] is aQ1-extension of V [K][U ] and 2ω ∩ V [K][U ][xn : n ∈ ω] = 2ω ∩ V [K][xn : n ∈ ω].Since (Fact 9.5.5) the poset Q1 adds no independent reals, a density argumentshows that U generates an ultrafilter in the model V [K][U ][xn : n ∈ ω].

Let A ⊂ ω × 2ω be the relation consisting of all pairs 〈n, x〉 such that x isE0-related to xn. In the model V [K][U ][xn : n ∈ ω] form the models Vn of allsets hereditarily definable from parameters xm for m ∈ n, A, and parametersin the model V [K][U ], and let M be the model of all sets hereditarily definablefrom the set A and its elements and some other parameters in V [K][U ]. Notethat by Vopenka’s theorem [45, page 249], the sequence 〈xn : n ∈ ω〉 is genericover the model M and the forcing adding it restores the axiom of choice in itsgeneric extension of M .

The following claim makes a critical use of the compact balance assumption.

Claim 9.5.7. In the model M :

1. there is a sequence 〈pn : n ∈ ω〉 such that pn is a balanced virtual conditionfor Vn, and p ≥ p0 ≥ p1 ≥ p2 ≥ . . . ;

2. any sequence as in (1) must have a lower bound.

Proof. Start with item (1). Working in V [K][U ], fix Rn-names χn so that Rnis the product of the first n-many copies of the Vitali forcing in the productQ1 and χn is a name for a virtual balanced condition stronger than p andthan all χm for m ∈ n. This is possible by Definition 9.2.1(2). Now, in themodel V [K][U ][xn : n ∈ ω] for each n ∈ ω let pn be the U -limit of the sequence〈χn/〈xim : m ∈ n〉 : i ∈ ω〉 in the compact space of virtual balanced conditionsin the model Vn, where xim ∈ 2ω is the binary sequence obtained from xm byreplacing its first i many entries with 0. The limit exists since U generates anultrafilter in all models Vn. The balanced virtual conditions pn for n ∈ ω forma decreasing sequence by Definition 9.2.1(3). Also, the definition of the limit


𝑞

𝑝0

𝑝1

V0 ��0

��1

��2

��3

V1

V2

V[H0] V[H1]

V3

Figure 9.2: A pod at work.


does not really depend on the sequence 〈xn : n ∈ ω〉 but only on the set A, andtherefore 〈pn : n ∈ ω〉 ∈M as desired.

Item (2) is somewhat trickier. Let N be some generic extension of M whichrestores the axiom of choice. In the model N , let B be the compact Hausdorffspace of balanced conditions for P . For each number n ∈ ω, consider theset Cn = {p ∈ B : p ≤ pn}. This is a nonempty and closed subset of B byDefinition 9.2.1(2) and (3). Since the closed sets Cn for n ∈ ω form a decreasingsequence, a compactness argument shows that there must be some p ∈

⋂n Cn.

This is the required lower bound of the sequence 〈pn : n ∈ ω〉.

Work in the model M . By the forcing theorem, there must be a poset R ofrank < κ restoring the axiom of choice, an R-name σ for a condition in the posetP stronger than all the balanced virtual conditions pn for n ∈ ω and an R-nameη for an element of ωω such that R Coll(ω,< κ) σ P τ([y]E) = [η]F forall functions y uniformizing the relation A. The following claim uses the podbalance assumption.

Claim 9.5.8. There is a condition r ∈ R and a point u ∈ ωω ∩M such thatr η F u.

Proof. Suppose towards a contradiction that this fails. Let H0, H1 ⊂ R be filtersmutually generic over the model M . Observe that the models M [H0],M [H1]are generic extensions of V mutually generic over the pod M : if a0 ∈ M [H0]and a1 ∈M [H1] are disjoint sets of ordinals, they must be separated by disjointsets b0, b1 of ordinals in the model M by the mutual genericity over M andProposition 1.7.8. Now, let p0 = σ/H0, u0 = η/H0, p1 = σ/H1 and u1 = η/H1.Note that for every condition p ∈ P ∩M , either both p0, p1 are below p or bothp0, p1 are incompatible with p. This occurs because by Claim 9.5.6, there is anumber n ∈ ω such that p ∈ Vn, the virtual condition pn is either below p orincompatible with it by its balance, and p0, p1 ≤ pn. By the pod balance ofthe poset P , the conditions p0, p1 have a lower bound in the poset P . By theforcing theorem in M and the initial contradictory assumption, u0, u1 are notF -related to any element of M . By Proposition 9.5.2, u0 cannot be F -relatedto u1. Thus, the lower bound of p0, p1 would have to force the unique E-classof all uniformizations of the set A to be mapped by τ simultaneously to [u0]Fand to [u1]F , which is impossible.

Now, the claim together with the forcing theorem means that there is a naturalnumber n ∈ ω, a point u ∈ Vn in ωω and in the model Vn a poset S of size< κ and an S-name χ for a condition in the poset P stronger than p and anS-name ξ for an element of (2ω)ω which is not E-related to any point in Vnsuch that S Coll(ω,< κ) χ P τ([ξ]E) = [u]F . Namely, the poset Sis a remainder of the iteration of Q and R after the first n-many Q-genericreals are added. Let H0, H1 ⊂ S be filters mutually generic over the modelVn = V [K][U ][xm : m ∈ n] and write p0 = χ/H0, y0 = ξ/H0, p1 = χ/H1, andy1 = ξ/H1. By the balance of the virtual condition pn, the conditions p0, p1 havea lower bound. Since the equivalence relation E is pinned, the points y0, y1 are


not E-related. By the forcing theorem, the lower bound of the conditions p0, p1

forces τ([y0]E) = τ([y1]E) = [u]F . This shows that τ cannot be an injection andcompletes the proof of the theorem.

Example 9.5.9. The poset P = P(ω) modulo finite is compactly balancedand pod balanced. Compact balance was proved in Example 9.2.4. Towards thepod balance, suppose that V [H0], V [H1] are generic extensions mutually genericover a pod and p0, p1 ∈ P are incompatible conditions in the respective models,i.e. almost disjoint subsets of ω. By the mutual genericity there must be almostdisjoint sets p′0, p

′1 ∈ V [H0] ∩ V [H1] of natural numbers such that p0 ⊂ p′0 and

p1 ⊂ p′1. These sets as conditions in P exemplify the pod balance.

Corollary 9.5.10. 1. Let P = P(ω) modulo finite. Then in the P -extensionof the symmetric Solovay model, |Eω0 | 6≤ |EKσ | holds.

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds, thereis a nonprincipal ultrafilter on ω and yet |Eω0 | 6≤ |EKσ |.

Example 9.5.11. Let K be a locally countable Borel simplicial complex ona Polish space X. Then the poset P = PK is pod balanced. To see this,let V [H0], V [H1] be models mutually generic over a pod and p0 ∈ V [H0] andp1 ∈ V [H1] are incompatible conditions. Write E = EK as in Definition 6.1.3;this is a countable Borel equivalence relation on X. The only way how p0, p1 canfail to be compatible is that there is an E-class c represented in both V [H0] andV [H1] and finite sets a0, a1 ⊂ b such that a0 ⊂ p0 and a1 ⊂ p1 and a0 ∪ a1 /∈ K.Use the Mostowski absoluteness between V [H0], V [H1] and V [H0, H1] to seethat c ⊂ V [H0] ∩ V [H1] holds. Thus, the conditions p′0 = a0 and p′1 = a1

witness the pod balance of the poset P .

Many posets associated with locally countable simplicial complexes are com-pactly balanced; we get for example the following.

Corollary 9.5.12. Let Γ be a Borel locally finite graph on a Polish space Xsatisfying the Hall’s marriage condition.

1. Let P be the poset adding a perfect matching to Γ with countable ap-proximations. Then in the P -extension of the symmetric Solovay model,|Eω0 | 6≤ |EKσ | holds.

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds, Γhas a perfect matching, and yet |Eω0 | 6≤ |EKσ |.

Example 9.5.13. Let P be the poset adding a linear ordering on the EKσ -quotient space as in Example 8.6.5. Then the poset P is compactly balancedand pod balanced. Compact balance was proved in Example 9.2.11. For thepod balance, suppose that V [H0], V [H1] are extensions mutually generic overa pod, and p0 ∈ V [H0], p1 ∈ V [H1] are incompatible conditions. Passing tostronger conditions if necessary, we may assume that p0, p1 are linear orders onsome subsets of the EKσ -quotient space. The only way how they can fail to be


compatible is that there are EKσ -classes c, d represented in both V [H0], V [H1]such that 〈c, d〉 ∈ p0 and 〈d, c〉 ∈ p1. In view of Proposition 9.5.2, c, d arerepresented in V [H0] ∩ V [H1] and so the conditions p′0 = {〈c, d〉} and p′1 ={〈d, c〉} belong to V [H0] ∩ V [H1] and exemplify the pod balance.

Corollary 9.5.14.

1. Let P be the EKσ -linearization poset. Then in the P -extension of thesymmetric Solovay model, |Eω0 | 6≤ |EKσ | holds.

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds, theEKσ -quotient space is linearly ordered and yet |Eω0 | 6≤ |EKσ |.

Example 9.5.15. Let Γ,∆ be abelian Polish groups, with ∆ divisible. LetP be the poset for adding a discontinuous homomorphism from ∆ to Γ as inDefinition 8.2.1. Then the poset P is compactly balanced and pod balanced.Compact balance was proved in Example 9.2.20. For the pod balance, supposethat V [H0], V [H1] are extensions mutually generic over a pod, and p0 ∈ V [H0],p1 ∈ V [H1] are incompatible conditions. Passing to stronger conditions if nec-essary, we may assume that Γ∩V [H0]∩V [H1] ⊂ dom(p0),dom(p1) holds. SinceΓ ∩ V [H0] ∩ V [H1] is a subgroup of Γ, the only way how p0, p1 can fail to becompatible is that there is a point γ ∈ V [H0] ∩ V [H1] with p0(γ) 6= p1(γ). LetO0, O1 ⊂ ∆ be basic open sets separating the values p0(γ) and p1(γ) and letA0 = {q ∈ P : γ ∈ dom(q) and q(x) ∈ O0} and A1 = {q ∈ P : γ ∈ dom(q) andq(x) ∈ O1}. Clearly, A0, A1 ⊂ P are analytic sets coded in V [H0] ∩ V [H1], thesums ΣA0,ΣA1 are incompatible in P , and p0 ≤ ΣA0 and p1 ≤ ΣA1 holds.

Corollary 9.5.16. Let Γ,∆ be abelian Polish groups, with ∆ divisible.

1. Let P be the homomorphism poset. Then in the P -extension of the sym-metric Solovay model, |Eω0 | 6≤ |EKσ | holds.

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds, thereis a discontinuous homomorphism from Γ to ∆ and yet |Eω0 | 6≤ |EKσ |.

This section leaves perhaps more questions open than it resolves. We contentourselves with quoting one glaring case:

Question 9.5.17. Does the conclusion of Theorem 9.5.4 stay in force if theassumption of pod balance on P is dropped?

9.6 The pinned divide

No unpinned equivalence relation can be reduced by a Borel function to a pinnedone, see Fact 2.3.2. This feature persists to the cardinality computations inbalanced extensions of the choiceless Solovay model, with a small proviso.

9.6. THE PINNED DIVIDE 237

Theorem 9.6.1. Let E,F be Borel equivalence relations on respective Polishspaces X,Y , E pinned and F unpinned. In a balanced extension of a symmetricSolovay model derived from an inaccessible limit of inaccessibles, |F | 6≤ |E|holds.

We do not know if the increase in the large cardinal strength of the large cardi-nal hypothesis is necessary. We do know though that one has to consider Suslinposets which are balanced everywhere below κ as opposed to just cofinally bal-anced. This is clear from Example 9.6.2 below.

Proof. Let κ be an inaccessible cardinal which is a limit of inaccessibles. Let Pbe a Suslin poset which is balanced below κ. Let W be the symmetric Solovaymodel derived from κ and work in the model W . Towards a contradiction, sup-pose that p ∈ P is a condition and τ a P -name for an injection from F -classesto E-classes. Both p, τ are definable from some parameter z ∈ 2ω and parame-ters in the ground model. The assumptions imply that there is an inaccessiblecardinal λ < κ of V and a filter K ⊂ Coll(ω,< λ) generic over V such thatz ∈ V [K].

Work in the model V [K]. The balanced assumption on the poset P impliesthat there is a balanced virtual condition p ≤ p in P . Theorem 2.7.1 shows thatE is pinned in V [K] and therefore has at most c = ℵ1 many virtual classes,while Theorem 2.5.11 shows that F has at least 2ℵ1 many virtual classes. Theargument splits into two cases:

Case 1. For every poset R of size < κ, every R-name for a condition σ ≤ p inthe poset P , every F -pinned R-name η for an element of X and every R-nameχ for an element of X such that R Coll(ω,< κ) σ P τ([η]F ) = [χ]E , thename χ is E-pinned.

In this case, by a counting argument with the virtual E- and F -classes, itmust be the case that there are posets R0, R1 and respective names σ0, η0, χ0

and σ1, η1, χ1 on them such that η0, η1 are F -pinned, χ0, χ1 are E-pinned, R0 Coll(ω,< κ) σ0 P τ0([η]F ) = [χ0]E and R1 Coll(ω,< κ) σ1 Pτ([η1]F ) = [χ1]E and 〈R0, χ0〉E〈R1, χ1〉 holds while 〈R0, η0〉F 〈R1, η1〉 fails. LetH0 ⊂ R0, H1 ⊂ R1 be filters generic over V [K], and write r0 = σ0/H0 ∈ P ,r1 = σ1/H1 ∈ P , x0 = χ0/H0 ∈ X, x1 = χ1/H1 ∈ X, and y0 = η/H0 ∈ Y andy1 = η1/H1 ∈ Y . The balance of the virtual condition p in P implies that r0, r1

are compatible in P with some lower bound r ∈ P , and the pinned assumptionsimply that x0 E x1 holds and y0 F y1 fails. Since W is the symmetric Solovayextension of both models V [K][H0] and V [K][H1], the forcing theorem in thesetwo models implies that in W , r P τ([y0)F ) = [x0]E and τ([y1]F ) = [x1]E ; thiscontradicts the assumption that τ is forced to be an injection.

Case 2. Case 1 fails. Then, there has to be poset R of size < κ, every R-namefor a condition σ ≤ p in the poset P , every F -pinned R-name η for an element ofX and every R-name χ for an element of X such that R Coll(ω,< κ) σ Pτ([η]F ) = [χ]E , such that the name χ is not E-pinned. In such a case, theremust be filters H0, H1 ⊂ R mutually generic over V [K] such that the pointsx0 = χ/H0, x1 = χ/H1 ∈ X are E-unrelated. Let r0 = σ/H0, r1 = σ/H1, and


let y0 = η/H0, y1 = η/H1 ∈ Y . The balance of the virtual condition p impliesthat the conditions r0, r1 ∈ P are compatible with a lower bound r, and thepinned assumptions imply that y0 F y1 holds. Since W is the symmetric Solovayextension of both models V [K][H0] and V [K][H1], the forcing theorem in thesetwo models implies that in W , r P τ([y0]F ) = [x0]E and τ([y1]F ) = [x1]E thiscontradicts the assumption that τ is forced to be a function.

Example 9.6.2. Let A ⊂ X = (2ω)ω be the set of all elements x ∈ X such thatrng(x) is linearly ordered by Turing reducibility, and let E = F � A. Note thatthe E-quotient space is classified by subsets of 2ω which are linearly ordered byTuring reducibility. There are uncountable sets of this form, and therefore E isunpinned. At the same time, such sets have size at most ℵ1 and so the virtualE-quotientspace has size 2ℵ1 .

Now, let P be the collapse of E = F2 � A to |2ω|. The poset P is balancedif and only if 2ℵ0 = 2ℵ1 by Theorem 6.6.3; in particular, it is cofinally balancedbelow any inaccessible cardinal. Thus, the P -extension of the Solovay modelis a cofinally balanced extension in which the (in ZFC) unpinned equivalencerelation E has the same cardinality as 2ω.

Chapter 10

Uniformization

The question whether various forms of uniformization hold in the models withinpurview of this book is one of the more slippery issues we set out to resolve.

10.1 Tethered Suslin forcing

In order to prove all forms of uniformization in a clean sweep, the followingdefinition will be central.

Definition 10.1.1. Let P be a Suslin forcing and λ be an infinite cardinal. Theposet P is λ-tethered if whenever V [H0], V [H1] are mutually generic extensionsof V and p0 ∈ V [H0], p1 ∈ V [H1] are incompatible conditions, then there areincompatible virtual conditions p′0, p

′1 ∈ V represented on posets of size < λ such

that p0, p′0 are compatible and p1, p

′1 are compatible. The poset P is tethered if

it is λ-tethered for some cardinal λ.

One immediate corollary of tether is that it places an upper bound on thenumber of balanced classes, which we do not know how to obtain in general.

Proposition 10.1.2. Let P be a Suslin forcing and λ be an infinite cardinal.

If P is λ-tethered then there are at most 22λ many balanced classes.

Proof. Let A be the set of all virtual conditions of P represented on posets of size< λ. It is a matter of elementary cardinal arithmetic to conclude that |A| ≤ 2λ.Whenever p is a balanced virtual condition, write Bp = {q ∈ A : p ≤ q}. Notethat for each q ∈ A, if q ∈ Bp then p ≤ q and if q /∈ Bp then p, q are incompatible.To prove the proposition, it will be enough to show that the set Bp characterizesthe balanced condition p.

Now suppose that p0, p1 are balanced virtual conditions such that Bp0 =Bp1 ; denote the common value by B and work to show that p0 = p1. To thisend, suppose that R0, R1 are arbitrary posets, H0 ⊂ R0 and H1 ⊂ R1 aremutually generic filters and p0 ∈ V [H0], p1 ∈ V [H1] are conditions below p0, p1

respectively. In view of the definition of the equivalence of balanced pairs,

239

240 CHAPTER 10. UNIFORMIZATION

it will be enough to show that p0, p1 are compatible. However, if they wereincompatible, by the tether assumption there would have to be incompatiblevirtual conditions p′0, p

′1 ∈ A such that p′0, p0 are compatible and p′1, p1 are

compatible. By the note from the previous paragraph, it must be the case thatp′0, p

′1 ∈ B holds. But then, p′0, p

′1 are not incompatible because p0, p1 are both

their common lower bounds.

10.2 Pinned uniformization

In this section, we show that sets whose vertical sections are E-classes for asuitably regular equivalence relation E can be uniformized.

Definition 10.2.1. Let E be an equivalence relation on a Polish space X. E-uniformization is the statement: if B ⊂ 2ω ×X is a set whose vertical sectionsare E-classes, then there is a function f ⊂ B such that for every y ∈ 2ω, ifBy 6= 0 then f(y) is defined (and is an element of By).

The central theorem of this section can now be stated using Convention 1.7.16.

Theorem 10.2.2. Let E be a pinned Borel equivalence relation on a Polishspace X. In tethered, cofinally balanced extensions of the symmetric Solovaymodel, E-uniformization holds.

To parse the statement of the theorem correctly, note that P needs to be tetheredin all forcing extensions and balanced only in cofinally many forcing extensionsbelow the inaccessible cardinal which gives rise to the symmetric Solovay model.

Proof. Let κ be an inaccessible cardinal. Let P be a Suslin poset which isbalanced cofinally below κ and tethered below κ. Let W be a symmetric Solovaymodel derived from κ. Let G ⊂ P be a filter generic over W and work in W [G].The following claim contains the key use of the tether assumption.

Claim 10.2.3. Let y ∈ W [G] be any set and let M be the class of all setshereditarily definable from G, y, and parameters in the ground model. Then Gcontains a realization of a balanced virtual condition in M .

A tacit part of the statement of the claim is that M is a model of ZFC, andtherefore it belongs to W by Theorem 9.1.1 by the cofinal balance assumptionon the poset P ; in fact, it is a generic extension of the ground model by a posetof size < κ.

Proof. Use the tether assumption to find a cardinal λ ∈ κ such that M |= Pis λ-tethered. Working in M , let D be the set of all virtual conditions in Prepresented on posets of size < λ. In W [G], let C ⊂ D be the set of all virtualconditions in the set D whose realization belongs to the generic filter G. Theset C has just been defined from the ordinal λ and the ultrafilter G, so C ∈M .By the genericity of the filter G, there must be a condition p ∈ G which is belowall conditions in C and incompatible with all the conditions in D \ C. By the

10.2. PINNED UNIFORMIZATION 241

forcing theorem, in M there have to be a poset Q of cardinality less than κ anda Q-name τ for a condition in P such that Q forces τ to be below all conditionsin C and incompatible with all conditions in D \ C, and such that there is afilter H ⊂ Q generic over the model M with p = τ/H. We claim that in themodel M , the pair 〈Q, τ〉 is balanced.

To see this, suppose towards contradiction that R0, R1 are posets and K0 ×H0 ⊂ R0 × Q and K1 × H1 ⊂ R1 × Q are filters mutually generic over themodel M and p0, p1 are conditions in the respective models below τ/H0 andτ/H1 respectively. We must show that p0, p1 are compatible. This, however,is immediately clear from the λ-tether assumption and the fact that for everyvirtual condition p on P represented on a poset of cardinality less than λ andfor every i ∈ 2, p ≥ pi ↔ p ∈ C and p is incompatible with pi if and only ifp /∈ C.

Still working in the model M , let p be a virtual balanced condition in thebalance class of 〈Q, σ〉, obtained from Theorem 5.2.5. Clearly, the filter Gcontains a realization of it.

Return to the model W for a moment. Let p ∈ G be a condition and let τ bea P -name for a subset of 2ω ×X such that each vertical section of τ is a singleE-class. Let z ∈ 2ω be a point such that both p, τ are definable from z and aground model parameter. Back in the model W [G], write B = τ/G. Let y ∈ 2ω

be an arbitrary point and let My be the model of all sets hereditarily definablefrom G, y, and parameters in the ground model. By Claim 10.2.3, there is abalanced virtual condition p in My whose realization belongs to the filter G.Necessarily, p ≤ p.

In the model My, we will show that Coll(ω,< κ) p P τy ∩My 6= 0. Oncethis is done, the set B can be uniformized by the set of all pairs 〈y, x〉 such thatx ∈My is the first point in the canonical well-ordering of the model My whichbelongs to By.

To prove the forcing statement in the previous paragraph, suppose towardsa contradiction that it fails. Move to the model My. There have to be aposet R of size < κ and R-names σ for a condition in P stronger than p andη for an element of X which has no E-equivalent in the model My such thatR Coll(ω,< κ) σ P 〈y, η〉 ∈ τ . In W , let H0, H1 ⊂ R be filters mutuallygeneric over the model My. Write p0 = σ/H0, p1 = σ/H1, x0 = η/H0 and x1 =η/H1. The balance of the condition p implies that p0, p1 ∈ P are compatibleconditions with a lower bound q. The pinned assumption on the equivalencerelation implies that x0, x1 ∈ X are not E-related.

Since W is also a symmetric Solovay extension of both My[H0] and My[H1],the forcing theorem applied in these two models implies that q x0, x1 ∈ τy.This is impossible as τ is forced to be an E-class and x0, x1 are not E-related.

Theorem 10.2.2 is the strongest possible result of its kind, as the followingobservation shows.

Theorem 10.2.4. Let E be an unpinned Borel equivalence relation on a Polishspace X. Then E-uniformization fails in balanced extensions of the Solovay


model.

Proof. Let W [G] be a balanced extension of the Solovay model. For each pa-rameter z ∈ 2ω the model Mz = HODV,z,G is well-ordered and therefore byTheorem 9.1.1, it is a well-ordered subclass of W and so an extension of V bya poset < κ. The Borel equivalence relation E is unpinned in Mz by Theo-rem 2.7.1. By Theorem 2.6.3, Mz |=there is a nontrivial E-pinned name on theposet Coll(ω, ω1). Note that (P(ω1))Mz is a countable set in W . Thus, onecan successfully define the set B ⊂ 2ω × X by setting 〈z, x〉 ∈ B if there is afilter g ⊂ Coll(ω, ωMz

1 ) generic over Mz such that τz/g E x, where τz is thefirst E-unpinned name in the canonical well-ordering of the model Mz. Everyvertical section of the set B then consists of precisely one E-equivalence class.

To see that the set B cannot be uniformized, note that every P -name in themodel W is definable from a real parameter and some additional parametersin the ground model, and therefore every element of W [G] is definable froma real parameter, the generic filter G, and some additional parameters in theground model. Thus, if f : 2ω → X is a putative uniformization of the set B,one can find a real parameter z ∈ 2ω such that f is definable from z,G and someelements of the ground model. Then f(z) should be an element of the modelMz by the definition of the model Mz. At the same time, the vertical sectionsBz contains no elements of Mz by the definition of the set B.

10.3 Well-orderable uniformization

In this section, we will prove that a strong version of the countable-to-one uni-formization statement holds in the extensions of the Solovay model by tetheredforcings.

Definition 10.3.1. Let E be an equivalence relation on a Polish space X.E-well-orderable uniformization is the statement: if E is a Borel equivalencerelation on a Polish space X and B ⊂ 2ω × X is a set, then the following areequivalent:

1. for every y ∈ 2ω, By is a union of a well-orderable collection of E-classes;

2. B =⋃αBα where for each ordinal α, Bα ⊂ 2ω ×X is a set whose verti-

cal sections are either empty or E-classes and the variable α ranges overordinals.

Clearly, only the implication (1)→(2) has nontrivial content. As a very specialcase, one can consider a set B ⊂ 2ω×X with all vertical sections countable andE the identity on X. Then (2) yields in particular a function uniformizing theset B: for each y with By nonempty find the least α = αy for which the verticalsection (Bα)y is nonempty and then let f(y) = x for the unique x ∈ X suchthat 〈y, x〉 ∈ Bαy . This is the familiar countable-to-one uniformization.

The central theorem of this section can now be stated using Convention 1.7.16.

10.3. WELL-ORDERABLE UNIFORMIZATION 243

Theorem 10.3.2. Let E be a Borel equivalence relation on a Polish space X.In tethered, cofinally balanced extensions of the symmetric Solovay model, E-well-orderable uniformization holds.

Proof. Let κ be an inaccessible cardinal. Let P be a Suslin poset which isbalanced cofinally below κ and tethered below κ. Let W the symmetric Solovaymodel derived from κ. In the model W , let p ∈ P be a condition and τ bea P -name such that p τ ⊂ 2ω × X is a set whose vertical sections are well-orderable unions of E-classes. Let z ∈ 2ω be a parameter such that p, τ are bothdefinable from z and some additional ground model parameters. Let G ⊂ P bea generic filter meeting the condition p and let B = τ/G. Work in the modelW [G].

Fix a point y ∈ 2ω and consider the model My of all sets hereditarily defin-able from parameters y, z,G and parameters in the ground model. Note thatMy is a model of ZFC and therefore finds itself already in the model W by The-orem 9.1.1; in fact, it is a generic extension of the ground model by a poset ofsize < κ. We claim that the vertical section By consists of realizations of virtualE-classes of the model My. Once this is proved, one can decompose the set Binto the union B =

⋃αBα by setting 〈y, x〉 ∈ Bα if 〈y, x〉 ∈ B and x belongs

to the realization of α-th virtual E-class in My in the canonical well-ordering ofthe model My.

To this end, use Claim 10.2.3 to argue that there is a balanced virtual con-dition p in the model My such that its realization belongs to the generic filterG. Necessarily p ≤ p must hold. Now, move to the model My and argue that inthis model, Coll(ω,< κ) p P τy consists of realizations of virtual E-classesof the model My. The proof of the theorem is then concluded by an appeal tothe forcing theorem applied in My.

Suppose towards a contradiction that the forcing statement in the previousparagraph fails and work in My. In view of Corollary 9.1.3, there exist a posetR0 of cardinality less than κ, an R0-name η for an element of Xω and an R0-name σ0 for an element of P stronger than p such that R0 Coll(ω,< κ) σ0 P ∀x ∈ X 〈y, x〉 ∈ η ↔ ∃i x E η(i). Also there must be a poset R1 ofsize < κ, an R1-name χ for an element of X which is forced not to realize anyvirtual E-class in My, and an R1-name σ1 for a condition in P stronger than psuch that R1 Coll(ω,< κ) σ1 〈y, χ〉 ∈ τ .

Working in W , let H0 ⊂ R0 and H1 ⊂ R1 be filters mutually generic overV , let u = η/H0 ∈ Xω, x = χ/H1 ∈ X, p0 = σ0/H0 ∈ P and p1 = σ1/H1 ∈ P .The balance of the condition p implies that p0, p1 are compatible conditions inthe poset P with a lower bound q. The choice of the name χ implies that x is notE-related to any element on the sequence u. Since W is a symmetric Solovayextension of the model My[H0], the forcing theorem applied in that model saysthat q τ = [rng(u)]E . Since W is a symmetric Solovay extension of the modelMy[H1], the forcing theorem applied in that model says that q x ∈ τ . Sincex /∈ [rng(u)]E , this is a contradiction.


10.4 Saint Raymond uniformization

In this section, we will prove that a strong version of Saint Raymond uniformiza-tion holds in generic extensions of the Solovay model by tethered balanced forc-ing.

Definition 10.4.1. Let I be a downward closed collection of closed subsets ofa Polish space X. I-Saint Raymond uniformization is the following statement:for every set B ⊂ 2ω ×X, the following are equivalent:

1. for every y ∈ 2ω, By is a union of a well-ordered collection of sets in I;

2. B =⋃αBα where for each ordinal α and each y ∈ 2ω, the vertical section

(Bα)y ⊂ X is closed and belongs to I.

As in the previous section, only the (1)→(2) implication has content. In the caseof I =the collection of singletons (plus the empty set), we can again derive thefamiliar countable-to-one uniformization as a very special case. The followingtheorem is stated using Convention 1.7.16.

Theorem 10.4.2. Let I be an analytic collection of closed subsets of a Polshspace X. In tethered, cofinally balanced extensions of the symmetric Solovaymodel, the I-Saint-Raymond uniformization holds.

Proof. Let κ be an inaccessible cardinal. Let P be a Suslin forcing which isbalanced cofinally below κ and tethered below κ. Let W be the symmetricSolovay model derived from κ and work in W . Let p ∈ P be a condition and τa P -name for a subset of 2ω×X whose vertical sections are well-ordered unionsof sets in I. Let z ∈ 2ω be a parameter such that p, τ are both definable fromz and some additional ground model parameters. Let G ⊂ P be a generic filtermeeting the condition p and let B = τ/G.

Work in W [G]. Fix a point y ∈ 2ω and consider the model My of setshereditarily definable from parameters G, z, y and parameters in the groundmodel. Note that My is a model of choice and therefore belongs to W already;in fact, it is an extension of the ground model by a poset of size < κ. We claimthat every nonempty vertical section By is a union of sets in I which are codedin the model My. Once this is proved, one can decompose the set B into theunion B =

⋃αBα by setting 〈y, x〉 ∈ Bα if 〈y, x〉 ∈ B and x belongs to the α-th

set in I in the model My in the canonical well-ordering of the model My, andthis α-th set is a subset of B.

To prove this, first use Claim 10.2.3 to find a balanced virtual condition pin My whose realization belongs to the generic filter G. Work in the model My

and argue that Coll(ω,< κ) p P τy is a union of a collection of ground modelcoded elements of I. By the forcing theorem applied in the model My, this willcomplete the proof.

To prove the forcing statement in the previous paragraph, work in My. Sup-pose towards a contradiction that it fails. By the assumptions on the name τ ,there must be a poset R0 of size < κ and R0-name η for a countable sequence

10.5. EXAMPLES 245

of elements of I and an R0-name σ0 for a condition in P stronger than p suchthat R0 Coll(ω,< κ) σ0 p 〈y, x〉 ∈ η ↔ ∃i x ∈ η(i). By the contradictoryassumption, there also must be a poset R1 of size < κ and R1-name χ for anelement of X and an R1-name σ1 for a condition in P stronger than p such thatR1 Coll(ω,< κ) σ1 〈y, χ〉 ∈ τ and χ does not belong to any element of Iwhich belongs to My and is a subset of τy.

Move to the model W . Let H0, H1 ⊂ R0, R1 be filters mutually generic overMy, and let p0 = σ0/H0 and p1 = σ1/H1. By the balance of the conditionp, p0, p1 ∈ P are compatible conditions with a lower bound q ∈ P . Let x =χ/H1 ∈ X. The contradiction is now reached by a split into cases.Case 1. There exist a condition r ∈ H1 and a closed set C ∈ rng(η/H0) suchthat the ground model closed set D = X \

⋃{O : O ⊂ X is a basic open set

such that r χ /∈ O} is a subset of C. By the closure of the collection I undersubsets, D ∈ I holds; by the definition of the set D, x ∈ D holds. Now, W is thesymmetric Solovay extension of both models My[H0] and My[H1]. The forcingtheorem applied in My[H0] shows that W |= q D ⊂ C ⊂ τ . The forcingtheorem applied in My[H1] shows that W |= q x ∈ D and x belongs to noground model closed set in I which is a subset of τy. Thus, the same conditionq forces two contradictory statements.Case 2. Case 1 fails. Then, by the mutual genericity of the filters H0, H1 itmust be the case that x /∈

⋃rng(η/H0). Now, W is the symmetric Solovay

extension of both models My[H0] and My[H1]. The forcing theorem applied inMy[H0] shows that W |= q τy =

⋃rng(η/H0). The forcing theorem applied

in My[H1] shows that W |= q x ∈ τy \⋃

rng(η/H0). So again, the samecondition q forces two contradictory statements. The proof is complete.

10.5 Examples

In this section we present several examples of tethered and untethered partialorders. In all affirmative examples below, the conclusion is that in view ofTheorems 10.2.2, 10.3.2, and 10.4.2, the extensions of the symmetric Solovaymodel by the posets in question satisfy the pinned uniformization, the well-orderable uniformization, and the Saint Raymond uniformization.

We start with simplicial complex posets which naturally live on Polish spacesas opposed to quotient spaces. These posets are ℵ0-tethered in all cases that wecan check.

Example 10.5.1. Let K be a Borel simplicial complex on a Polish space X ofBorel coloring number ℵ1. Then the poset P = PK is ℵ0-tethered.

Proof. Let ~Γ be a Borel orientation of the graph Γ = ΓK of Definition 6.1.2in which every node gets countable outflow. Suppose that V [H0], V [H1] aremutually generic extensions of V and p0 ∈ V [H0], p1 ∈ V [H1] are incompatibleconditions. There must be finite sets a0 ⊂ p0 and a1 ∈ p1 such that a0∪a1 /∈ K;choose a0, a1 to be inclusion-minimal. Then they are connected in the graph Γ;assume for definiteness that a1 is in the ~Γ-outflow of a0.


Since the ~Γ-outflow of a0 is countable, Mostowski absoluteness shows that itis a subset of V [H0]; in particular, a1 ∈ V [H0]. By the product forcing theorem,a1 ∈ V must hold. Let A0 = {q ∈ P : there is a set c ⊂ q such that a1 belongs

to the ~Γ-outflow of c}, and A1 = {q ∈ P : a1 ⊂ C}. The sets A0 and A1 areanalytic subsets of P coded in V , ΣA0 and ΣA1 are incompatible in P , andp0 ≤ ΣA0, p1 ≤ ΣA1 holds.

Example 10.5.2. Let K be a modular Borel simplicial complex on a Polishspace X. Then the poset P = PK is ℵ0-tethered.

Proof. Let Y be a Polish space and f : K → [Y ]ℵ0 be a Borel function witnessingthe modularity of K. Suppose that V [H0], V [H1] are mutually generic extensionsof V and p0 ∈ V [H0], p1 ∈ V [H1] are incompatible conditions. There must befinite sets a0 ⊂ p0 and a1 ∈ p1 such that a0 ∪ a1 /∈ K. Write b = a0 ∩ a1 andobserve that f(a0) ∩ f(a1) 6= f(b) by the modularity of the function f .

Let O0, O1 ⊂ X be basic open sets separating the finite disjoint sets a0 \ b,a1 \ b. Use the mutual genericity to argue that b ∈ V and f(a0) ∩ f(a1) ⊂ V .Find a point x ∈ f(a0) ∩ f(a1) which is not in f(b), and let A0 = {q ∈ P :there is a set c ⊂ q such that b ⊂ c, c \ b ⊂ O0, and x ∈ f(c)}, and similarlyA1 = {q ∈ P : there is a set c ⊂ q such that b ⊂ c, c\b ⊂ O1, and x ∈ f(c)}. Thesets A0, A1 are analytic subsets of P coded in V , ΣA0 and ΣA1 are incompatiblein P by the modularity of the function f , and p0 ≤ ΣA0, p1 ≤ ΣA1 holds.

Example 10.5.3. Let K be a Gδ-matroid on a Polish space X as in Defini-tion 6.5.1. Then the poset P = PK is ℵ0-tethered.

Proof. Suppose that V [H0], V [H1] are mutually generic extensions of V andp0 ∈ V [H0], p1 ∈ V [H1] are incompatible conditions. Expanding the modelsV [H0], V [H1] and strengthening the conditions p0, p1 if necessary, we may as-sume that for every point x ∈ X ∩ V , either x ∈ p0 or p0 ∪ {x} is not a K-set,and similarly for p1.

Let q0 = p0 ∩ V and q1 = p1 ∩ V . There are several cases. If q0 6= q1,with say a point x ∈ q0 \ q1, then let A0 = {q ∈ P : x ∈ q} and A1 = {q ∈P : ∃a ∈ [q]<ℵ0 a ∪ {x} /∈ K}. It is clear that both sets A0, A1 ⊂ P are analyticand their suprema ΣA0 and ΣA1 are incompatible in P , and p0 ≤ Σp′0 andp1 ≤ Σp′1. Suppose then that q0 = q1, denote the common value p and usethe product forcing theorem to show that p ∈ V holds. Suppose first that pis not a maximal K-set. Then there must be x ∈ X \ p such that p ∪ {x}is a K-set and inclusion-minimal finite sets a0 ⊂ p0 and a1 ⊂ p1 such thata0 ∪ {x} /∈ K and a1 ∪ {x} /∈ K. By the assumption on x, a0 6⊂ V and a1 6⊂ Vmust hold. Let b = a0 ∩ a1 and let O0, O1 ⊂ X be basic open sets separatingthe nonempty sets a0 \b and a1 \b; note that b ∈ V holds by the product forcingtheorem. Let A0 = {q ∈ P : b ⊂ q and there is a finite set c ⊂ q ∩O0 such thatb ∪ c ∪ {x} /∈ K} and A1 = {q ∈ P : b ⊂ q and there is a finite set c ⊂ q ∩ O0

such that b ∪ c ∪ {x} /∈ K}. Clearly, A0, A1 ⊂ P are analytic sets coded in theground model, ΣA0 and ΣA1 are incompatible in P by the exchange propertyof the matroid K, and p0 ≤ ΣA0 and p1 ≤ ΣA1.

10.5. EXAMPLES 247

We come to the last case, where p is a maximal K-set in the ground model.This is impossible though: p is then a balanced virtual condition in P by The-orem 6.5.2, p0 ≤ p and p1 ≤ p, and so p0, p1 must be compatible by the balanceof p. This contradicts the initial choice of p0, p1.

Example 10.5.4. Let Γ be a Borel graph on a Polish space X of countablecoloring number. The Γ-coloring poset P of Definition 8.1.1 is ℵ0-tethered.

Proof. We identify P with its Suslin dense subset isolated in Theorem 8.1.2.Suppose that V [H0], V [H1] are mutually generic extensions of V and p0 ∈ V [H0],p1 ∈ V [H1] are incompatible conditions. Expanding the models V [H0], V [H1]and strengthening the conditions p0, p1 if necessary, we may assume that forevery point x ∈ X ∩ V , x ∈ dom(p0) and x ∈ dom(p1) both hold.

Let q0 = p0 � V and q1 = p1 � V . Suppose first that q0 6= q1. Then theremust be a point x ∈ V and distinct numbers n0, n1 ∈ ω such that q0(x) = n0

and q1(x) = n1. Let A0 = {q ∈ P : x ∈ dom(q) ∧ q(x) = n0} and A1 = {q ∈P : x ∈ dom(q) ∧ q(x) = n1}. Clearly, A0, A1 ⊂ P are analytic sets coded inthe ground model, ΣA0 and ΣA1 are incompatible in P and p0 ≤ ΣA0 andp1 ≤ ΣA1.

Now suppose that q0 = q1 and work towards a contradiction. Write p for thecommon value and note that by the product forcing theorem, p ∈ V holds. Nowp is a balanced virtual condition in P by Theorem 8.1.2, p0 ≤ p and p1 ≤ p, andso p0, p1 must be compatible by the balance of p. This contradicts the initialchoice of p0, p1.

Example 10.5.5. Let Γ,∆ be abelian Polish groups, with ∆ divisible. Thehomomorphism poset of Definition 8.2.1 is ℵ0-tethered.

Proof. Suppose that V [H0], V [H1] are mutually generic extensions of V andp0 ∈ V [H0], p1 ∈ V [H1] are incompatible conditions. Expanding the modelsV [H0], V [H1] and strengthening the conditions p0, p1 if necessary, we may as-sume that for every point γ ∈ Γ ∩ V , γ ∈ dom(p0) and Γ ∈ dom(p1) bothhold.

Let q0 = p0 � V and q1 = p1 � V . Suppose first that q0 6= q1. Then theremust be a point γ ∈ V and disjoint basic open sets O0, O1 ⊂ ∆ such thatp0(γ) ∈ O0 and p1(γ) ∈ O1. Let A0 = {q ∈ P : γ ∈ dom(q) ∧ q(γ) ∈ O0} andA1 = {q ∈ P : γ ∈ dom(q) ∧ q(γ) = O1}. Clearly, A0, A1 ⊂ P are analytic setscoded in the ground model, ΣA0 and ΣA1 are incompatible in P and p0 ≤ ΣA0

and p1 ≤ ΣA1.

Now suppose that q0 = q1 and work towards a contradiction. Write p for thecommon value and note that by the product forcing theorem, p ∈ V holds. Nowp is a balanced virtual condition in P by Theorem 8.2.2, p0 ≤ p and p1 ≤ p, andso p0, p1 must be compatible by the balance of p. This contradicts the initialchoice of p0, p1.

The following two examples deal with quotient simplicial complex posets.


Example 10.5.6. Let E ⊂ F be Borel equivalence relations on a Polish spaceX such that every F -class consists of countably many E-classes. The E,F -transversal poset P of Definition 6.6.5 is tethered.

Proof. We will show that P is λ-tethered where λ = iω1. Let V [H0], V [H1] be

mutually generic extensions of the ground model V , and let p0 ∈ V [H0] andp1 ∈ V [H1] be incompatible conditions in P . There must be elements x0 ∈ p0

and x1 ∈ p1 which are F -related but not E-related. In view of Proposition 2.1.7,the points x0, x1 must both be realizations of a virtual F -class in V and byClaim 6.6.13 there are virtual E-classes d0 6= d1 in V such that x0 is a realizationof d0 and x1 is a realization of d1.

In V , let p′0 = {q ∈ P : q contains some realization of the virtual E-class d0}and p′1 = {q ∈ P : q contains some realization of the virtual E-class d1}. Theseare virtual conditions on the posets which carry the virtual E-classes d0, d1, andthese are of size < λ by Theorem 2.5.6. In addition, p′0, p

′1 are incompatible,

and p0 ≤ p′0 and p1 ≤ p′1 as required.

The following example covers all posets which associate a structure to each G-class where G is a countable Borel equivalence relation on a Polish space, as inExample 6.6.10.

Example 10.5.7. Let X be a Polish space and let E ⊂ F be Borel equivalencerelations on X such that F is pinned and for each F -equivalence class C ⊂ X,E � C is smooth. Then the E,F -transversal poset P of Definition 6.6.5 isℵ0-tethered.

Proof. Let V [H0], V [H1] be mutually generic extensions of the ground model V ,and let p0 ∈ V [H0] and p1 ∈ V [H1] be incompatible conditions in P . There mustbe elements x0 ∈ p0 and x1 ∈ p1 which are F -related but not E-related. Sincethe equivalence relation F is pinned, there must be a point x ∈ X ∩ V whichis F -related to both x0, x1. By the initial assumptions on E,F , in the groundmodel there also must be a Borel map h : [x]F → 2ω reducing E to the identity.Since h(x0) 6= h(x1), there must be distinct binary strings s0, s1 ∈ 2n for somen ∈ ω such that s0 ⊂ h(x0) and s1 ⊂ h(x1). Let A0 = {q ∈ P : p ∩ h−1[s0] 6= 0}and A1 = {q ∈ P : p ∩ h−1[s1] 6= 0}. It is clear that A0, A1 are analytic subsetsof P coded in the ground model, ΣA0 is incompatible with ΣA1 in P , andp0 ≤ ΣA0 and p1 ≤ ΣA1 as required.

Example 10.5.8. Let F be a Fraisse class in a finite relational language withstrong amalgamation. Let E be a Borel equivalence relation on a Polish spaceX. Then the E,F-Fraisse poset P of Definition 8.6.3 is tethered. In addition,if E is pinned, then P is ℵ0-tethered.

Proof. Start with the general case. Let λ = iω1 . We will show that the posetP is λ-tethered. For brevity, assume that F has just one relational symbol ofits language of some finite arity n ∈ ω. Thus, a condition p ∈ P is just asingle n-ary relation on a countable set dom(p) which respects the equivalenceE, and such that its E-quotient is a F-structure. Let V [H0], V [H1] be mutually

10.5. EXAMPLES 249

generic extensions of the ground model V and p0 ∈ V [H0] and p1 ∈ V [H1] beincompatible conditions. In view of the strong amalgamation assumption, theremust be n-tuples ~x0 and ~x1 in dom(p0) and dom(p1) respectively such that foreach i ∈ n, ~x0(i) E ~x1(i) holds, and ~x0 ∈ p0 and ~x1 /∈ p1 (or vice versa).

By Proposition 2.1.7, there is an n-tuple ~d of virtual E-classes in V such thatfor each i ∈ n, ~x0(i), ~x1(i) are realizations of ~d(i). Let p′0 = {q ∈ P : dom(q)

contains an n-tuple ~x which is a realization of ~d and ~x ∈ q} and p′1 = {q ∈P : dom(q) contains an n-tuple ~x which is a realization of ~d and ~x /∈ p}. Theseare virtual conditions on the posets which carry the virtual E-classes on thetuple ~d, and these are of size < λ by Theorem 2.5.6. In addition, p′0, p

′1 are

incompatible, and p0 ≤ p′0 and p1 ≤ p′1 as required.The case of a pinned equivalence relation E proceeds in the same way, noting

that in this case theE-quotient space and the virtual E-quotient space coincide.

The ultrafilter posets typically do satisfy the tether demands.

Example 10.5.9. The poset P of all infinite subsets of ω ordered by inclusionis ℵ0-tethered.

Proof. Let V [H0], V [H1] be mutually generic extensions of V , containing the re-spective conditions p0, p1 ∈ P which are incompatible in P . Removing finitelymany numbers from each, we may assume that in fact p0 ∩ p1 = 0. By Propo-sition 1.7.8, there are disjoint sets p′0, p

′1 ⊂ ω in the ground model such that

p0 ⊂ p′0 and p1 ⊂ p′1. These sets as conditions in P exemplify the tether of theposet P .

Example 10.5.10. Let A be a Ramsey sequence of finite structures. The posetP = PA of Definition 7.3.2 is ℵ0-tethered.

Proof. We use the terminology of Section 7.3. In particular, the sequence A iswritten as 〈An : n ∈ ω〉 of structures on pairwise disjoint sets. Let n ∈ ω be anatural number. Write Dn for the set of all copies of An which are a subset ofsome Am for m ≥ n.

Let V [H0], V [H1] be mutually generic extensions of the ground model V , andlet p0 ∈ V [H0] and p1 ∈ V [H1] be incompatible conditions in P . Expanding themodels and strengthening the conditions if necessary, by Claim 7.3.5 we mayassume that for each n ∈ ω and each set b ⊂ Dn in the ground model, eitherpAn0 ⊂ b or pAn0 ∩ b = 0 modulo finite, and similarly for subscript 1. Let (Fn)0

be the set of all ground model elements of P(Dn) for which the first optionprevails, and similarly for subscript 1.

Suppose first that there is a number n ∈ ω for which (Fn)0 6= (Fn)1. Fordefiniteness, suppose that there is some set b ∈ (Fn)0 \ (Fn)1. Then let p′0 ={p ∈ P : pAn ⊂ b modulo finite} and p′1 = {p ∈ P : pAn ∩ b = 0 modulo finite}.The sets p′0, p

′1 ⊂ P are analytic and coded in the ground model. Also, clearly

Σp′0,Σp′1 are incompatible in P , and p0 ≤ Σp′0 and p1 ≤ Σp′1 as desired.

Now suppose that the sets (Fn)0, (Fn)1 are equal for all n ∈ ω. Write Fnfor their common value. The product forcing theorem shows that Fn ∈ V and


even 〈Fn : n ∈ ω〉 ∈ V . It is not difficult to check that it is an A-sequence ofultrafilters. Theorem 7.3.4 then shows that it yields a virtual balanced conditionp, and p0, p1 ≤ p. Since p0, p1 ≤ p, the balance of p shows that p0, p1 arecompatible conditions, contradicting the initial choice of p0, p1.

Example 10.5.11. Let 〈Γ, ·〉 be a countable semigroup. The poset P = P (Γ)of Section 7.4 is ℵ0-tethered.

Proof. We adopt the terminology of Section 7.4. Let V [H0], V [H1] be mutuallygeneric extensions of V , containing the respective conditions p0, p1 ∈ P whichare incompatible in P . By Proposition 7.4.3, extending the models and strength-ening the conditions if necessary, we may assume that both p0, p1 sort out theset P(ω)∩V . Let q0 = {a ⊂ Γ: a ∈ V and p0 accepts a} and q1 = {a ⊂ Γ: a ∈ Vand p1 accepts a}.

Suppose first that q0 6= q1, and for definiteness assume that there is a seta ∈ q0 \ a1. Let p′0 = {q ∈ P : q accepts a} and p′1 = {q ∈ P : q declines a}. It isclear that p′0, p

′1 ⊂ P are analytic sets coded in V , Σp′0,Σp

′1 are incompatible,

and p0 ≤ Σp′0, p1 ≤ Σp′1.Suppose now that q0 = q1 and work towards a contradiction. Denote the

common value by p and use the product forcing theorem to see that p ∈ V . Sincethe conditions p0, p1 sort out P(Γ)∩V , p is an ultrafilter. In V , Proposition 7.4.5then shows that p is an idempotent ultrafilter, and by Theorem 7.4.7, it yields abalanced condition in the poset P . Since p0, p1 ≤ p, the balance of p shows thatp0, p1 are compatible conditions, contradicting the initial choice of p0, p1.

Finally, we include two examples of balanced posets which are not tethered.

Example 10.5.12. Let E be a non-smooth Borel pinned equivalence relationon a Polish space X. The collapse poset P of |E| to 2ω of Definition 6.6.2 isnot tethered. Countable-to-one uniformization fails in the resulting extensionof the symmetric Solovay model.

Proof. We prove the last sentence. The poset P is balanced by the pinnedassumption on E and Corollary 6.6.4. In the P -extension of the symmetricSolovay model, |E| ≤ |2ω| holds by the definition of the poset P . In addition,|E0| ≤ |E| holds by the Glimm–Effros dichotomy as E is assumed to be non-smooth. In sum, |E0| ≤ |2ω| holds, as witnessed by some function g : 2ω → 2ω.In addition, Corollary 11.6.3 shows that in the P -extension of the symmetricSolovay model, E0 has no transversal, and therefore the function g has no leftinverse. This feature of g stands witness to the failure of the countable-to-oneuniformization.

Example 10.5.13. Let X be an uncountable Polish space. The Kurepa poset Pof Definition 8.4.2 is not tethered. In the P -extension of the symmetric Solovaymodel, countable-to-one uniformization fails.

Proof. For simplicity of notation, assume that X = 2ω. Let W [F ] be the sym-metric Solovay model derived from an inaccessible cardinal κ, with the generic

10.5. EXAMPLES 251

Kurepa family F attached. In W [F ], let E be the equivalence relation on Xconnecting x, y if there is no set in F containing x but not y or vice versa. Allclasses of the equivalence relation E are countable as the family F consists ofcountable sets and is cofinal. We will show that in W [F ] there is no total func-tion f : X → X such that for every x ∈ X, if [x]E 6= {x} then f(x) ∈ [x]E \ {x}.This is clearly in violation of countable-to-one uniformization.

Working in W , let p ∈ P be a condition and τ be a P -name such thatp τ : X → X is a function. We must find x ∈ X and a stronger conditionp ≤ p which forces that [x]E 6= {x} and τ(x) /∈ [x]E \ {x}. The condition p andthe name τ are definable from some parameter z ∈ 2ω and some ground modelparameters. Let V [K] be an intermediate model obtained by a forcing of size< κ which contains the parameter z. Work in the model V [K].

Let Q be the Cohen forcing on X, adding the generic point xgen . LetH0, H1 ⊂ Q be mutually generic filters, let x0 = xgen/H0 and x1 = xgen/H1,and in the model V [K][H0][H1] consider the virtual condition p ≤ p which inaddition to p contains the sets

⋃p ∪ {x0} ∪ [x1]E0

and X ∩ V [K][H0][H1]. ByTheorem 8.4.3, p is a balanced virtual condition. By a balance argument, itdecides the value of τ(x0) to be some specific point y ∈ X. An inspection ofthe condition p reveals that it forces [x0]E = {x0} ∪ [x1]E0 . We will be finishedif we show that y is not E0-related to x1.

Towards contradiction, assume that it is. Then y is a Cohen generic pointover V [K][H0], yielding the same extension as x1. By the forcing theorem inV [K][H0] and the definability of τ , there would have to be a condition q ∈ Qsuch that V [K][H0] |= q Coll(ω,< κ) p τ(x0) = xgen . By the forcingtheorem again, in the model V [K][H0][H1], the virtual condition p would forcethe value τ(x0) to be each of the infinitely many elements of [x1]E0

in the openset q simultaneously, an impossibility.

There are also partial orders where we are unable to decide the status of tether,such as the Fin×Fin poset of Definition 7.2.1.


Chapter 11

Locally countable structures

This chapter introduces a methodology for proving that certain models of ZF+DCwe produce do not contain locally countable structures of certain type. In partic-ular, we learn how to produce models of ZF+DC which contain locally countablestructures of one type but not of another type. The whole enterprise should beviewed as a parallel to the extensive field of descriptive set theory of locallycountable structures. There are many striking similarities present, and manyothers are sure to be found in the future.

11.1 Central objects and notions

In this section, we introduce certain basic locally countable graphs and hy-pergraphs and the associated concerns of the descriptive set theory of locallycountable structures. Some of the critical issues deal with the comparison ofchromatic and Borel chromatic numbers of various analytic hypergraphs on Pol-ish spaces. An analytic hypergraph on a Polish space X is a set G ⊂ [X]<ℵ0

which is an analytic subset of the hyperspace K(X) with the Vietoris topology.The elements of a hypergraph are referred to as hyperedges. Thus, our hyper-graphs contain finite hyperedges only–they are finitary ; it may occur thoughthat there is no fixed n ∈ ω (arity) such that G ⊂ [X]n. As a matter of con-vention, our hypergraphs contain no singleton sets. A graph is a hypergraph ofarity 2. If G,H are hypergraphs on respectve spaces X,Y , a function h : X → Yis a homomorphism if the h-image of any G-hyperedge is an H-hyperedge. Ananticlique is a set a ⊂ X such that [a]<ℵ0 ∩G = 0. The chromatic number of Gis the smallest cardinality of a set of anticliques covering the whole space X; inthe choiceless context, where cardinalities are not well-ordered, we distinguishonly between countable and uncountable chromatic number and different valuesof countable chromatic numbers.

Most of the central analytic hypergraphs in this section are in fact productsof hypergraphs on finite or countable domain. The following definition explainsthe typical construction.

253

254 CHAPTER 11. LOCALLY COUNTABLE STRUCTURES

Definition 11.1.1. Let 〈an, Hn, tn : n ∈ ω〉 be a sequence such that an is anonempty countable (often finite) set, Hn is a hypergraph on an and tn ∈∏m∈n am are finite strings such that {tn : n ∈ ω} is a dense subset of

⋃n

∏m∈n am.

Let X =∏n an and define:

1. (the skew product)∏nHn, tn is the hypergraph on X containing all sets

b ⊂ X such that for some n ∈ ω, ∀m 6= n ∀x0, x1 ∈ b x0(m) = x1(m),∀x ∈ b tn ⊂ b, and {x(n) : x ∈ b} ∈ Hn;

2. (the product)∏nHn is the hypergraph on X containing all sets b ⊂ X

such that for some n ∈ ω, ∀m 6= n ∀x0, x1 ∈ b x0(m) = x1(m), and{x(n) : x ∈ b} ∈ Hn.

In all cases, the sets an are implicit in the notation. The definition of theproduct (as opposed to the skew product) does not depend on the strings tn.Among the product graphs, the Hamming graphs are prominent.

Definition 11.1.2. Let n ≥ 2 be a natural number. Hn, the Hamming graphon nω, is the product of infinitely many cliques of size n. H<ω, the diagonalHamming graph on the space

∏n(n+ 1), is the product of cliques of all possible

nonzero finite cardinalities. Finally, Hω, the Hamming graph on ωω, is theproduct of infinitely many cliques on ω.

Skew products are useful as minimal examples of various phenomena. Onestandard example is the uncountable Borel chromatic number of graphs:

Definition 11.1.3. Let G be an analytic hypergraph on a Polish space X. Ghas countable Borel chromatic number if there are Borel G-anticliques Bn ⊂ Xfor n ∈ ω such that X =

⋃nBn.

Definition 11.1.4. For each n ∈ ω, let an = 2, let Hn be the graph on ancontaining only the whole set an as an edge and let tn ∈ 2n be a string suchthat {tn : n ∈ ω} ⊂ 2<ω is dense. The graph G0 is defined as

∏nHn, tn.

Fact 11.1.5. [54](ZF+DC) For every analytic graph G on a Polish space X,exactly one of the following occurs:

1. the Borel chromatic number of G is countable;

2. there is a continuous map from 2ω to X which is a homomorphism of G0

to G.

Uncountable chromatic numbers of locally countable graphs can be used inZF+DC to rule out seemingly unrelated phenomena. The following humbleproposition will be useful at several points of this book:

Proposition 11.1.6. (ZF+DC) If the chromatic number of G0 is greater thantwo then

1. there is no linear ordering of the E0-quotient space;

11.1. CENTRAL OBJECTS AND NOTIONS 255

2. |E0| 6≤ |2ω|;

3. there is no discontinuous homomorphism between Polish groups.

Proof. For the first item, consider the equivalence relation E on 2ω connectingpoints x, y ∈ 2ω if they differ in finite, even number of entries. Then E ⊂ E0; asa subset of a hyperfinite equivalence relation, it is itself hyperfinite and thereforeBorel reducible to E0 [48, Theorem 8.1.1]. Now, suppose that the E0-quotientspace is linearly orderable; then so is the E-quotient space, and one can definethe function c : 2ω → 2 by setting c(x) = 0 if among the two E-classes whichconstitute the E0-class of x, x belongs to the smaller one in the fixed linearordering of the E-quotient space. It is not difficult to show that c is a G0-coloring with two colors.

The second item follows from the first, since the inequality |E0| ≤ |2ω| yieldsa linear ordering on the E0-space by simply pulling back the usual lexicograph-ical order on 2ω to the E0-quotient space via the assumed injection from theE0-quotient space to 2ω. The third item is much harder. [76] constructs, inZF+DC, a coloring of the Hamming graph H2 (a superset of the G0-graph)with two colors from the assumption that there is a discontinuous homomor-phism between Polish groups.

It is not difficult to extend the G0-dichotomy to arbitrary finitary analytic hy-pergraphs. In this generality, one has to allow for uncountably many basishypergraphs, which are nevertheless easy to describe:

Definition 11.1.7. Suppose that 〈an, Hn, tn : n ∈ ω〉 is a sequence such thateach an is a a finite set of cardinality at least two, Hn is the hypergraph on itcontaining an as its only hyperedge, and tn ∈

∏m∈n am is a string such that

the set {tn : n ∈ ω} is dense in⋃n

∏m∈n am. Then the hypergraph

∏nHn, tn

on∏n an is called a principal skew product.

The following can be proved in the same way as the G0-dichotomy. We suppressthe standard proof.

Fact 11.1.8. (ZF+DC) Let G be a finitary analytic hypergraph on a Polishspace X. Exactly one of the following occurs:

1. the Borel chromatic number of G is countable;

2. there exists a principal skew product and a continuous homomorphism ofit to the hypergraph G.

In this chapter, we will also use more involved variations of the uncountableBorel chromatic number.

Definition 11.1.9. Let G be an analytic finitary hypergraph on a Polish spaceX. The hypergraph G has Borel σ-bounded chromatic number if there are Borelsets Bn ⊂ X for n ∈ ω such that X =

⋃nBn and for every n ∈ ω, every finite

subset of Bn has G-chromatic number less than n+ 2.


It is immediate that countable Borel chromatic number implies Borel σ-boundedchromatic number. The opposite implication does not holds as the followingexamples show.

Example 11.1.10. The Hamming graph H2 has uncountable Borel chromaticnumber by Fact 11.1.5 but it does have Borel σ-bounded chromatic number.To see this, observe that it contains no odd length cycles and therefore H2 oneach finite subset of 2ω has chromatic number 2. The sequence Bn defined byB1 = 2ω and Bn = 0 for all n 6= 1 exemplifies the Borel σ-bounded chromaticnumber of H2.

Example 11.1.11. The diagonal Hamming graph H<ω does not have Borelσ-bounded chromatic number. To see this, by the Baire category theorem it isenough to show that any non-meager Borel subset of X =

∏n(n + 1) contains

arbitrarily large finite H<ω-cliques. Thus, let B ⊂ X be non-meager Borel andlet m ∈ ω be a number. Let t be a finite string of natural numbers such that Bis comeager in t and |t| = n > m. A simple construction yields a point x ∈ Xsuch that t ⊂ x and for every i ∈ n the point xi, obtained from x by rewritingits n-th entry with i, belongs to the set B. Since the set {xi : i ∈ n} ⊂ B is anH<ω-clique, the argument is complete.

The class of analytic finitary hypergraphs which do not have Borel σ-boundedchromatic number has a simple basis.

Definition 11.1.12. Let 〈an, Hn, tn : n ∈ ω〉 be a sequence such that an isa nonempty finite set, Hn is a hypergraph on an of chromatic number > n,tn ∈

∏m∈n am, and the set {tn : n ∈ ω} is dense in

⋃n

∏m∈n an. The skew

product∏nHn, tn is then called a large skew product.

A straightforward Baire category argument as in Example 11.1.11 shows thata large skew product does not have Borel σ-bounded chromatic number. Thefollowing fact is proved in the same way as the G0-dichotomy, and we omit thestandard argument.

Fact 11.1.13. Let G be a finitary analytic hypergraph on a Polish space X.Exactly one of the following occurs:

1. G has Borel σ-bounded chromatic number;

2. there is a large skew product H on a Polish space Y and a continuoushomomorphism h : Y → X of H to G.

Many interesting natural examples of hypergraphs which do not have Borel σ-bounded chromatic number actually have a stronger property encapsulated inthe following definitions.

Definition 11.1.14. Let a be a finite set and G be a hypergraph on a. Thefractional chromatic number of G is the maximum of all numbers Σv∈af(v)where f : a → [0, 1] ranges over all functions such that Σv∈bf(v) ≤ 1 for allG-anticliques b ⊂ a.


A word about the terminology is in order. In fractional graph theory [78], thefractional chromatic number is typically defined for graphs only. In the graphcontext, the definition above corresponds to the fractional clique number of G,which is equal to the standard fractional chromatic number of G by a linearprogramming duality argument [78, Section 3.1]. In order to maintain coherentterminology, we neglect this important but for our ends irrelevant point. Atrivial restatement of the fractional chromatic number will be used below: it isthe largest real number r such that there is a probability measure on a in whichevery G-anticlique has mass ≤ 1/r.

Note that the fractional chromatic number is no greater than the chromaticnumber. Just as in the case of the chromatic number, if G0, G1 are hypergraphson a0, a1 respectively and h : a0 → a1 is a homomorphism of G0 to G1 thenthe fractional chromatic number of G1 is not smaller than the fractional chro-matic number of G0. The following examples further elucidate the relationshipbetween the chromatic number and its fractional counterpart.

Example 11.1.15. For every n ∈ ω let Gn be the graph on [n]2 connectingpairs a, b if the smaller element of a is equal to the larger element of b or viceversa. Then the chromatic numbers of Gn tend to infinity with n, but thefractional chromatic numbers remain bounded by 4.

For the former statement, note that if n is such that n → (3)2m then the

chromatic number of Gn is greater than m: if [n]2 =⋃i∈mBi, then by the

Ramsey property of n there is i ∈ m and a triple c ⊂ n such that [c]2 ⊂ Bi;however, [c]2 clearly contains a G-edge.

For the latter statement, suppose that µ is a probability measure on [n]2,and let λ be the normalized counting measure on P(n). Let B = {〈a, x〉 ∈[n]2 × P(n) : x contains the smaller number in a but not the larger number ina}. The vertical sections of B have λ-mass 1/4, so by the Fubini theorem theremust be a horizontal section Bx whose µ-mass is ≥ 1/4. It is not difficult tocheck that Bx is a G-anticlique.

Example 11.1.16. Let ε > 0 be a fixed real number. For each n ∈ ω, letGn be the graph on 2n which connects x, y if the set {m ∈ n : x(m) = y(m)}has size at most εn. The fractional chromatic numbers of Gn tend to infinitywith n. To see this, consider the normalized counting measure µn on 2n, thenormalized Hamming metric dn on 2n, and the concentration of measure resultsfor the Hamming cubes [73, Theorem 4.3.19]: for every δ > 0 there is a numberk ∈ ω such that for every n ≥ k and every set b ⊂ 2n of µn-mass greater thanδ, the ε-neighborhood of b in 2n in the sense of the metric dn has mass greaterthan 1/2. For each such number n ∈ ω, every set b ⊂ 2n of µn-mass greaterthan δ contains a Gn-edge. To produce the edge, consider the automorphismπn : 2n → 2n defined by πn(x)(m) = 1− x(m). The set π′′b has µ-mass greaterthan δ again. The ε/2-neighborhoods of the sets b and π′′b in the sense ofthe metric dn have both µn mass greater than 1/2 and therefore intersect. Itfollows that there must be points x ∈ b and y ∈ π′′b with dn(x, y) ≤ εn, andthen x, π−1y are Gn-connected elements of the set b.


Example 11.1.17. For every n ∈ ω, let Hn be the hypergraph on n of arity3 consisting of all arithmetic progressions of length 3. Then the fractionalchromatic numbers of Hn tend to infinity with n–just consider the normalizedcounting measure on n and the density van der Waerden’s theorem [88].

Definition 11.1.18. Let G be an analytic finitary hypergraph on a Polish spaceX. We say that G has Borel σ-bounded fractional chromatic number if there areBorel sets Bn ⊂ X such that X =

⋃nBn and for each n ∈ ω, the hypergraph

G has fractional chromatic number less than n+ 2 on every finite set a ⊂ Bn.

Clearly, Borel σ-bounded chromatic number implies Borel σ-bounded fractionalchromatic number but the opposite implication fails. The most natural class ofBorel graphs exemplifying the distinction is the following:

Example 11.1.19. Let E be a Borel non-smooth equivalence relation on aPolish space X. Let Q be the poset of all pairs q = 〈aq, bq〉 of finite subsets ofX such that (aq × bq) ∩ E = 0; the ordering is that of coordinatewise reverseinclusion. Let G be the graph connecting two conditions in Q if they are incom-patible. Then G does not have Borel σ-bounded chromatic number, yet it doeshave Borel σ-bounded fractional chromatic number.

Proof. To show that G does not have Borel σ-bounded chromatic number, firstuse the Glimm–Effros dichotomy to find a Borel reduction h : 2ω → X of E0 toE. For E0-unrelated points y0, y1 ∈ 2ω let r(y0, y1) = 〈{h(y0)}, {h(y1)}〉 andnote that r(y0, y1) ∈ Q. By the Baire category theorem, it will be enough, forany Borel nonmeager set B ⊂ 2ω × 2ω and every n ∈ ω, to produce a finitesubset b ⊂ B such that the chromatic number of G on q′′b is greater than n.To this end, let t0, t1 ∈ 2<ω be two binary strings of the same length such thatB ⊂ [t0] × [t1] is comeager. Let k ∈ ω be a number such that k → (3)2

n. Asimple construction yields points yi ∈ 2ω for i ∈ k such that for any i ∈ j ∈ k,〈ta0 yi, t

a1 yj〉 ∈ B holds. It will be enough to show that the graph G on the set

c = {q(ta0 yi, ta1 yj) : i ∈ j ∈ k} has chromatic number greater than n. To see this,

suppose that c =⋃m∈n dm. By the choice of the number k, there exist m ∈ n

and numbers i0 ∈ i1 ∈ i2 such that any pair of them gives rise to a conditionin the set dm. Now note that the conditions r(ta0 yi0 , t

a1 yi1) and r(ta0 yi1 , t

a1 yi2)

are incompatible in Q and so form a G-edge.To show that G has Borel σ-bounded fractional chromatic number, let Bn =

{q ∈ Q : |aq ∪ bq| ≤ n}, observe that the set Bn is Borel and Q =⋃nBn, and

argue that for every finite set c ⊂ Bn, the fractional chromatic number on G on cis ≤ 2−n. Indeed, assume that µ is any probability measure on the set c; we willproduce a set of µ-mass ≥ 2−n consisting of pairwise compatible conditions, i.e.a G-anticlique. To do this, let e be the finite set of all E-classes with nonemptyintersection with

⋃q∈c(aq ∪ bq) and let λ be the normalized counting measure

on P(e). Let A = {〈q, u〉 ∈ c × P(e) : aq ⊂⋃u ∧ bq ∩

⋃u = 0}. Since c ⊂ Bn,

all vertical sections of the set A have λ-mass ≥ 2−n. By the Fubini theorem,there is a horizontal section of A of µ-mass ≥ 2−n. This horizontal section isthe desired G-anticlique of µ-mass ≥ 2−n.


Example 11.1.11 actually shows that the diagonal Hamming graph does not haveBorel σ-bounded fractional chromatic number, since the density of a clique ofsize n is n. Many other examples arise from various density versions of Ramsey-type theorems.

Example 11.1.20. Suppose that Z acts in a Borel, free and measure-preservingway on a Polish probability measure space 〈X,µ〉. Let G be the Borel hyper-graph on X of arity 3 containing a triple {x, y, z} if there is n ∈ ω such thatn · x = y and n · y = z. The hypergraph G does not have Borel σ-boundedfractional chromatic number.

Proof. Suppose towards a contradiction that X =⋃nBn is a decomposition of

X into Borel sets witnessing the Borel σ-bounded fractonal chromatic number ofG. Pick n ∈ ω and ε > 0 such that µ(Bn) > ε. Use the density version of van derWaerden theorem [88] to find a number m0 ∈ ω such that in every subset of m0

of size > m0/n, one of the classes contains an arithmetic progression of lengththree. Use the density van der Waerden theorem again to find a number m1

such that every subset of m1 of size > εm1 contains an arithmetic progressionof length m0. Let C ⊂ m1 × X be the set C = {〈i, x〉 : x ∈ i · Bn}. As everyvertical section of the set C has µ-mass > ε, by the Fubini theorem there mustbe a point x ∈ X such that the horizontal section Cx ⊂ m1 contains more thanεm1 many numbers and so an arithmetic progression {i+ jk : k ∈ m0} for somechoice of i, k. Let xj = (−i−jk) ·x for j ∈ m1 and note that {xj : j ∈ m0} ⊂ Bnholds. By the choice of m0, the fractional chromatic number of G on the set{xj : j ∈ m0} is larger than n, contradicting the choice of the set Bn.

The class of analytic finitary hypergraphs which do not have Borel σ-boundedfractional chromatic number has a simple basis.

Definition 11.1.21. Suppose that 〈an, Hn, tn : n ∈ ω〉 is a sequence such thatan is a nonempty finite set, Hn is a hypergraph on an such that every Hn-anticlique has fewer than |an|/(n + 2) many elements, and tn ∈

∏m∈n am and

the set {tn : n ∈ ω} is dense in⋃n

∏m∈n an. The skew product

∏nHn, tn is

then called a large measured skew product.

A straightforward Baire category argument shows that a large measured skewproduct does not have Borel σ-bounded fractional chromatic number. The fol-lowing fact is proved in the same way as the G0-dichotomy, and we omit thestandard argument.

Fact 11.1.22. Let G be a finitary analytic hypergraph on a Polish space X.Exactly one of the following occurs:

1. G has Borel σ-bounded fractional chromatic number;

2. there is a large measured skew product H on a Polish space Y and acontinuous homomorphism h : Y → X of H to G.

We now include two definitions and related dichotomies which deal with theclique number of Borel graphs.


Definition 11.1.23. Let G be an analytic graph on a Polish space X. We saythat G has Borel σ-bounded clique number if there are Borel sets Bn ⊂ X forn ∈ ω such that no Bn contains a G-clique of size n+ 2.

Fact 11.1.24. Let G be an analytic graph on a Polish space X. Exactly one ofthe following occurs:

1. G has Borel σ-bounded clique number;

2. there is a continuous homomorphism from a skew product of infinitelymany cliques of increasing finite size to G.

Definition 11.1.25. Let G be an analytic graph on a Polish space X. We saythat G has Borel σ-finite clique number if there are Borel sets Bn ⊂ X for n ∈ ωsuch that no Bn contains an infinite G-clique.

Fact 11.1.26. Let G be an analytic graph on a Polish space X. Exactly one ofthe following occurs:

1. G has Borel σ-finite clique number;

2. there is a continuous homomorphism from a skew product of infinitelymany infinite cliques to G.

11.2 Very Suslin forcings

The preservation arguments in this chapter are in spirit quite different than thosein other chapters. All of them use definable c.c.c. control forcings. These areposets that serve to build interesting balanced conditions in the central σ-closedposet. We need to learn how to iterate the control forcings with finite supportwhile preserving their definability properties. We also need to isolate somedefinable regularity properties of the control forcings and learn how to preservethose. This last task is quite reminiscent of Stevo Todorcevic’s emphasis onregularity properties of c.c.c. posets as exhibited in numerous papers of his–[93,92]. However, we must stress that in our case the regularity properties must bewitnessed in a definable way, otherwise they are worthless. Similarly, our c.c.c.control forcings are nearly all isomorphic to the product of continuum manyCohen reals in ZFC; their worth for our purpose stems from their definabilityproperties as opposed to their forcing properties.

Definition 11.2.1. [44] A pre-ordering 〈P,≤〉 is a Suslin forcing if there is anambient Polish space X such that P ⊂ X is an analytic set, ≤ is an analyticsubset of X2, and the incompatibility relation ⊥ on P is an analytic subset ofX2. To ease the notational clutter, in this section we only require that ≤ isa transitive relation on P containing the diagonal. In addition, the poset P isrequired to have a largest element.

11.2. VERY SUSLIN FORCINGS 261

A given Suslin poset 〈P,≤〉 can be reinterpreted in every forcing extension.Note that the demands on the analytic definitions of ≤ and ⊥ are Π1

2 and sothe reinterpretation is again a Suslin poset. Importantly, the c.c.c. property ofSuslin forcings persists to forcing extensions.

Fact 11.2.2. [44] Let 〈P,≤〉 be a Suslin c.c.c. poset. The reinterpretation of Pin any forcing extension is c.c.c.

One has to enter the much more restrictive class of very Suslin forcings to guar-antee that the finite support iterations, separative quotients, and completionshave desirable descriptive properties.

Definition 11.2.3. Suppose that P is a Suslin c.c.c. poset on an ambient Polishspace X. The poset is very Suslin if the set {a ∈ Pω : rng(a) is predense in P}is an analytic subset of Xω.

Note that being a very Suslin partial order is a Π12 statement in the code for

the analytic set of maximal antichains and therefore absolute among all forcingextensions. Note that if the poset P and the ordering ≤ are Borel, which isthe case for all applications in this book and elsewhere as well, then both theincompatibility relation and the set of maximal antichains are coanalytic andtherefore Borel by the Suslin theorem.

Example 11.2.4. The Cohen forcing, random forcing, and the eventually dif-ferent real forcing are very Suslin.

Proof. The Suslinity of the definition of the posets is elementary and left for thereader to check; we only verify the very Suslin property. The Cohen forcing isthe poset P of finite binary strings ordered by reverse inclusion. A set a ⊂ P ispredense if every condition in P is compatible with some element of a, which inview of the fact that P is countable is a Borel condition. If the random forcingP is realized as the collection of compact µ-positive mass subsets of X orderedby inclusion, where µ is some Borel probability measure on a Polish space X,then a countable set a ⊂ P is pre-dense just in case µ(

⋃a) = 1, which is a Borel

condition. The case of the eventually different real forcing is somewhat moreinvolved, and addressed in [101, Proposition 3.8.12].

Example 11.2.5. The Hechler forcing is Suslin c.c.c. but not very Suslin.

The Suslinness and c.c.c. of Hechler forcing are well-known and easily checked.The failure of the the very Suslin property follows from the basic distinctionbetween Suslin and very Suslin c.c.c. posets: the latter cannot add dominatingreals.

Proposition 11.2.6. Let P be a very Suslin c.c.c. forcing. Then P does notadd dominating reals.

Proof. Let p ∈ P and τ be a P -name for an element of ωω. Consider the setA = {T ⊂ ω<ω : some q ≤ p forces T to have no infinite branch modulo finite


dominated by τ}. By the evaluation of the complexity of the forcing relationin Proposition 11.2.9 below, A is an analytic set of trees. At the same time, ifp forces τ ∈ ωω to be a dominating real, then A is the set of all well-foundedtrees, which is coanalytic and not analytic. Since the conclusion is impossible,the assumption must be false as well. The proposition follows.

The main feature of the class of very Suslin c.c.c. posets is that it is in a suitableprecise sense closed under finite support iterations of countable length.

Theorem 11.2.7. Let α ∈ ω1 be a countable ordinal and P a very Suslin c.c.c.forcing. The finite support iteration of P of length α is naturally isomorphic toa very Suslin c.c.c. forcing.

We emphasise that no such iteration result is possible for general Suslin forcings.The complexity of the forcing relation on e.g. the Hechler forcing causes thecomplexity of the iterated posets to explode in ways that are unmanageablewithout large cardinal assumptions. In order to prove the iteration theorem forvery Suslin c.c.c. posets succintly, we need several preliminary definitions andfacts.

Definition 11.2.8. Suppose that P is a very Suslin c.c.c. poset and X is aPolish space.

1. A layered P -name for an element of X is a tuple τ = 〈An, fn : n ∈ ω〉 suchthat An ⊂ P is a maximal antichain, fn is a map from An to basic opensubsets of X of radius < 2−n, and if m < n then Am refines An and ifpn ∈ An and pm ∈ Am are such that pm ≤ pn, then the closure of fm(pm)is a subset of fn(pn).

2. XP is the set of all layered P -names for elements of X.

Note that the set XP is analytic in a suitable ambient space. The definition ofXP appears to depend on the choice of the metric for X, a dependence that wehappily suppress. It is clear that for every name for an element of the space Xthere is a layered name which is forced to be equal to the original name. Nowwe are ready to show that the forcing relation of very Suslin c.c.c. forcings isΣ1

1 on Σ11 in a precise sense.

Proposition 11.2.9. Suppose that P is a very Suslin c.c.c. poset, X is a Polishspace, and A ⊂ X is an analytic set. The set {〈p, τ〉 ∈ P ×XP : p τ ∈ A} isan analytic subset of P ×XP .

Proof. Let τ = 〈An, fn : n ∈ ω〉. Let h : ωω → A be a continuous surjection. Thestatement p τ ∈ A is equivalent to the existence of a system 〈Bn, gn : n ∈ ω〉where for each n ∈ ω, Bn is a maximal antichain of P of conditions which areeither below p or incompatible with p, gn : Bn → ω<ω, Bn+1 refines Bn andAn, whenever r ≤ q are conditions in Bn+1 and Bn respectively then gn+1(r)properly extends gn(q), and if r ≤ q is an element of Bn+1 below p and anelement of An respectively then [gn+1(r)] ⊂ h−1fn(q). This is an analyticstatement.

11.2. VERY SUSLIN FORCINGS 263

In order to formulate the finite support iterations of very Suslin forcings, we muststate the usual two-step iteration and direct limit definitions in the definablecontext. These definitions are verbose but contain no surprises.

Definition 11.2.10. Let P,Q be posets. A projection of Q to P is a pair oforder-preserving functions π : Q→ P and ξ : P → Q such that

1. π ◦ ξ is the identity on P ;

2. whenever π(q) ≤ p then q ≤ ξ(p);

3. whenever p ≤ π(q) then there is q′ ≤ q such that π(q′) ≤ p.

The following definition will be used at the successor stage of the iterations.

Definition 11.2.11. Let P be a very Suslin poset on a Polish space X andQ be a very Suslin poset on a Polish space Y . Define P ∗ Q to be the set ofall ordered pairs 〈p, q〉 where p ∈ P and q is a nice P -name for an elementof Y such that p q ∈ Q. The ordering is defined by 〈p1, q1〉 ≤ 〈p0, q0〉 ifp1 ≤ p0 and p1 q1 ≤ q0. The iteration P ∗ Q also includes the iteration mapsπ : P ∗ Q→ P defined by π(p, q) = p together with the function ξ : P → P ∗ Qdefined by ξ(p) = 〈p, 1Q〉.

The limit stages will be defined in the following way.

Definition 11.2.12. A very Suslin system is a tuple 〈Pn : n ∈ ω, πnm, ξmn : m ≤n ∈ ω〉 where

1. each Pn is a very Suslin c.c.c. forcing;

2. for each m ≤ n ∈ ω, the functions πnm : Pn → Pm and ξmn : Pm → Pnare analytic and form a projection of Pn to Pm, with πnn and ξnn equalto the identity on Pn;

3. the functions πnm commute, as do the functions ξmn.

The limit of the system is the poset Pω of all pairs 〈p, n〉 where p ∈ Pn, orderedby 〈q, n〉 ≤ 〈p,m〉 if m ≤ n and πnm(q) ≤ p in Pn. The limit also includes thelimit maps πωm : Pω → Pm and ξmω : Pm → Pω defined by ξmω(p) = 〈p,m〉 andπωm(〈p, n〉) = πnm(p) if n > m and πωm(〈p, n〉) = ξnm(p) if n ≤ m.

Finally, we are ready to approach the proof of Theorem 11.2.7.

Proof of Theorem 11.2.7. The argument proceeds by a standard transfinite in-duction argument using the following claims:

Claim 11.2.13. Let P,Q be very Suslin c.c.c. posets. Then

1. P ∗ Q is a very Suslin c.c.c. poset;

2. the iteration maps are analytic and form a projection from P ∗ Q to P .


Proof. This is a routine complexity calculation. The underlying set P ∗ Q isanalytic by Proposition 11.2.9, since p q ∈ Q is an analytic statement. Theordering on P ∗ Q is analytic for the same reason. To check the analyticityof the incompatibility relation, observe that conditions 〈p0, q0〉 and 〈p1, q1〉 areincompatible just in case there is a (countable) maximal antichain A ⊂ P suchthat for every p ∈ A, either p is incompatible with p0, or it is incompatible withp1, or it is below both p0, p1, and in the latter case, p q0, q1 are incompatiblein the poset Q. This is an analytic statement by Proposition 11.2.9 and theassumption that the incompatibility relation on Q is analytic.

The poset P ∗ Q is c.c.c. because Q remains c.c.c. in the P -extension byFact 11.2.2 and so P ∗ Q is an iteration of two c.c.c. forcings. Finally, wehave to check that for a countable set B = {〈pn, qn〉 : n ∈ ω} ⊂ P ∗ Q, thestatement that B is predense is analytic. To see this, let Z be the Polishspace resulting from adding an isolated point 0 to Y , and consider the followingformula φ: there exists a name τ for an element of Zω and maximal antichainsAn ⊂ P for n ∈ ω such that for each n, An consists of elements which are eitherincompatible with pn or stronger than pn, if they are stronger than pn then theyforce τ(n) = qn, if they are incompatible with pn then they force τ(n) = 0, and1P forces rng(τ) ∩ Y is predense in Q. Parsing the formula φ, we see that itis analytic by Proposition 11.2.9, and that it says that 1P {qn : pn belongsto the generic filter} is predense in Q, which is exactly equivalent to the set Bbeing predense in P ∗ Q.

Checking the properties of the projection functions is routine and left to thereader.

Claim 11.2.14. Suppose that 〈Pn : n ∈ ω, πnm, ξmn : m ≤ n ∈ ω〉 is a verySuslin system. The limit Pω is a very Suslin c.c.c. forcing. The limit maps πωmand ξmω form analytic projections and commute with the projection maps of thevery Suslin system.

Proof. This is a straightforward complexity computation. Let Xn be the ambi-ent Polish space of each poset Pn; the ambient space of the poset Pω is then thedisjoint union

⋃nXn in product with ω. The ordering of Pω is analytic by the

definition. The incompatibility relation is analytic as well: suppose that 〈q, n〉and 〈p,m〉 are conditions in Pω, say with n ≥ m. These two conditions areincompatible just in case πnm(q) is incompatible with p in Pm by the definitoryproperties of projections.

The poset Pω is c.c.c. since it is a direct limit of c.c.c. forcings. To checkthe very Suslin property of the poset Pω, note that a countable set A ⊂ Pω ispredense if and only if for every number n ∈ ω, there is a maximal antichainBn ⊂ Pω of conditions such that for each p ∈ Bn, either there is a condition〈q,m〉 ∈ A such that m ≤ n and πnm(p) ≤ q in Pm, or there is a condition〈q,m〉 ∈ A such that m ≥ n and p ≤ πmn(q) in Pn. This is an analyticstatement as all the posets Pn are very Suslin.

Checking the projection properties of the functions πωm and ξmω is routineand left to the reader.

11.3. ITERATION THEOREMS 265

Theorem 11.2.7 follows.

The operation of the finite support product appears to be more difficult tohandle. Here the main question remains open:

Question 11.2.15. The class of Suslin c.c.c. forcings is closed under countable,finite support product. Is this true for the class of very Suslin c.c.c. forcings aswell?

11.3 Iteration theorems

The posets we use later in this chapter are fairly innocent from ZFC point ofview. However, we need their regularity properties to be witnessed in a Suslinway, and to be preserved under finite support iterations of countable length.This section contains a number of rather routine, but still apparently novel,regularity properties and their associated preservation theorems.


1. A set A ⊂ P is linked if any two conditions in A have a common lowerbound;

2. P is Suslin σ-linked if P can be written as a countable union⋃nAn of

linked analytic sets.

Note that the definitory properties of the cover P =⋃nAn are Π1

2 and thereforepersist to all forcing extensions. The main general result of this section is theiteration preservation theorem for Suslin σ-linkedness:

Theorem 11.3.2. Let P be a very Suslin c.c.c. forcing which is Suslin σ-linked.Let α ∈ ω1 be a countable ordinal. Then the finite support iteration of P of lengthα is a Suslin σ-linked forcing.

Proof. The argument proceeds by a straightforward transfinite induction argu-ment given the following two claims.

Claim 11.3.3. Let P,Q be very Suslin c.c.c. forcings, both of which are Suslinσ-linked. The P ∗ Q is Suslin σ-linked.

Proof. Let P =⋃nAn and Q =

⋃mBm be the covers of P,Q by analytic

centered (or linked) sets. Let Cnm ⊂ P ∗Q be the set of all conditions 〈p, q〉 suchthat there exists a condition 〈p′, q′〉 ≤ 〈p, q〉 such that p′ ∈ An and p′ q′ ∈ Bm.It is not difficult to check that P ∗ Q =

⋃nm Cnm, the sets Cnm are linked.

Moreover, the sets Cnm are analytic by Proposition 11.2.9.

Claim 11.3.4. Let 〈Pn : n ∈ ω, πnm, ξmn : m ≤ n ∈ ω〉 be a very Suslin systemconsisting of Suslin σ-linked forcings. Then the limit is Suslin σ-linked.


Proof. Let Pω be the limit of the system as described in Definition 11.2.12. LetPn =

⋃mAnm : n ∈ ω be a cover by analytic linked sets for each n ∈ ω. For

each n,m ∈ ω let Bnm ⊂ Pω be the set {〈p, n〉 ∈ Pω : p ∈ Anm}. It is immediatethat these are linked analytic sets covering Pω.

The theorem follows.

Theorem 11.3.5. The finite support product of countably many very Suslin,Suslin σ-linked posets, if very Suslin, is Suslin σ-linked.

Proof. Let Pn for n ∈ ω be the very Suslin posets, with their Suslin σ-linkedproperty witnessed by sets Anm for m ∈ ω. Consider the finite support productQ =

∏n Pn and assume it is very Suslin. For each number n0 ∈ ω and each

function h : n0 → ω let Bh = {q ∈ Q : dom(q) = n ∧ ∀m ∈ n0 q(m) ∈ Am,h(m)}.It is not difficult to verify that the sets Bh witness the Suslin σ-linked propertyof the poset Q.


1. A set A ⊂ P is centered if any finite subset of A has a lower bound in P ;

2. The poset P is Suslin σ-centered if P can be written as a countable union⋃nAn and the sets An are analytic and centered.

The key feature of the Suslin σ-centered property is that it is preserved underthe finite support iterations of very Suslin forcings. The proof of the followingtheorems are literally copied from Theorems 11.3.2 and 11.3.5, replacing theword “linked” with “centered”.

Theorem 11.3.7. Let P be a very Suslin c.c.c. forcing which is Suslin σ-centered. Let α ∈ ω1 be a countable ordinal. Then the finite support iterationof P of length α is a Suslin σ-centered forcing.

Theorem 11.3.8. The finite support product of countably many very Suslin,Suslin σ-centered posets, if very Suslin, is Suslin σ-centered.

Next comes a regularity property of Suslin forcing notions more permissive thancenteredness or linkedness, which should be compared to [86, Definition 3].


1. A set A ⊂ P is Ramsey-centered if for every number r ∈ ω there isk ∈ ω such that every k-tuple (with possible repetitions) of elements of Acontains an r-tuple with a common lower bound.

2. P is Suslin σ-Ramsey-centered if there is a cover P =⋃nAn by analytic

Ramsey-centered sets.

Note that the definitory properties of the covers are Π12 and therefore persist

to all forcing extensions. The most important result of this section is thatthe Suslin Ramsey-centeredness is a property preserved under finite supportiterations of very Suslin forcings.


Theorem 11.3.10. Let P be a very Suslin c.c.c. forcing which is Suslin Ramsey-centered. Let α ∈ ω1 be a countable ordinal. Then the finite support iterationof P of length α is a Suslin Ramsey-centered forcing.

Proof. The argument proceeds by a straightforward transfinite induction argu-ment given the following two claims.

Claim 11.3.11. Let P,Q be very Suslin c.c.c. forcings, both of which are SuslinRamsey-centered. The P ∗ Q is Suslin Ramsey-centered.

Proof. Find a cover Q =⋃mBm by analytic Ramsey-centered sets and a cover

P =⋃nAn by analytic Ramsey-centered sets. Let Cnm ⊂ P ∗ Q be the set of

all conditions 〈p, q〉 such that there exists a condition 〈p′, q′〉 ≤ 〈p, q〉 such thatp′ ∈ Anm and p′ q′ ∈ Bm. It is not difficult to check that P ∗ Q =

⋃nm Cnm,

the sets Cnm are Ramsey-centered: given r ∈ ω there is a k ∈ ω such that everyk-tuple in the set Bm contains an r-tuple with a common lower bound, andthere is an l ∈ ω such that every l-tuple in the set An contains a k-tuple witha common lower bound. Then, every l-tuple of conditions in Cmn contains anr-tuple with a common lower bound. Moreover, the sets Cnm are analytic byProposition 11.2.9.

Claim 11.3.12. Let 〈Pn : n ∈ ω, πnm, ξmn : m ≤ n ∈ ω〉 be a very Suslin systemconsisting of Suslin Ramsey-centered forcings. Then the limit is Suslin Ramsey-centered.

Proof. Let Pω be the limit of the system as described in Definition 11.2.12. LetPn =

⋃mAnm : n ∈ ω be a cover by analytic Ramsey-centered sets for each

n ∈ ω. For each n,m ∈ ω let Bnm ⊂ Pω be the set {〈p, n〉 ∈ Pω : p ∈ Anm}. Itis immediate that these are Ramsey-centered analytic sets covering Pω.

The theorem follows.

Theorem 11.3.13. The finite support product of countably many very Suslin,Suslin σ-Ramsey-centered posets, if very Suslin, is Suslin σ-Ramsey-centered.

Proof. Let Pn for n ∈ ω be the very Suslin posets, with their Suslin Ramsey-centered property witnessed by sets Anm ⊂ Pn for m ∈ ω. Consider the finitesupport product Q =

∏n Pn and assume it is very Suslin. For each function h ∈

ω<ω let Bh = {q ∈ Q : dom(q) = dom(h) ∧ ∀m ∈ dom(q) q(m) ∈ Am,h(m). Weproceed to verify that the sets Bh witness the Suslin Ramsey-centered propertyof the poset Q.

Fix the function h and let r ∈ ω be a number. By recursion on n ∈ dom(h)find numbers kn ∈ ω so that every k(0)-tuple of elements of A0,h(0) of size k(0)contains an r-tuple with a common lower bound in P0, and every k(n+ 1)-tupleof elements of An+1,h(n+1) contains a k(n)-tuple with a common lower bound inPn+1. Let n = max(dom(h)) and k = k(n). It is easy to see that every k-tupleof elements of Bh contains an r-tuple with a lower bound in the poset Q.


The existence of finitely additive measures on complete Boolean algebras is auseful tool in forcing theory, see [47]. The Suslin version of it needs to reflect thefact that finitely additive measures on uncountable sets are typically not Suslinin any sense. We use the following poor man’s version of a finitely additivemeasure:

Definition 11.3.14. Let P be a very Suslin c.c.c. poset. We say that P isSuslin measured if there are analytic sets An ⊂ P and positive rationals εn > 0such that P =

⋃nAn and for every n ∈ ω and every finite sequence 〈pi : i ∈ j〉

of elements of An with possible repetitions there is a set b ⊂ j of size at leastεnj such that the set {pi : i ∈ b} has a lower bound in P .

Note that the definitory properties of the cover P =⋃nAn are Π1

2 and thereforepersist to all forcing extensions. Note also that every Suslin measured poset isSuslin Ramsey-centered as witnessed by the same analytic sets.

Theorem 11.3.15. Let P be a very Suslin c.c.c. poset. If P is Suslin measured,then so are all of its finite support iterations of countable length.

Proof. As in the previous subsections, we proceed with claims dealing with thesuccessor and limit stage of the iteration.

Claim 11.3.16. Suppose that P,Q are very Suslin, c.c.c. posets which areSuslin measured. Then P ∗ Q is Suslin measured.

Proof. Let An, αn for n ∈ ω and Bn, βn for n ∈ ω witness the Suslin measuredproperty of P,Q respectively. Let Cnm ⊂ P ∗ Q be the upward closure of theset of all pairs 〈p, q〉 such that p ∈ An and p q ∈ Bm. Let γnm = αnβm.We claim that the sets Cnm and the numbers γnm witness the Suslin measuredproperty of the iteration.

It is clear from Proposition 11.2.9 that the sets Cnm are analytic; theyalso clearly exhaust all of P ∗ Q. Now, fix numbers n,m ∈ ω. Suppose that〈〈pi, qi〉 : i ∈ j〉 is a sequence of conditions in the set Cnm; strengthening theconditions we may assume that pi ∈ An and pi qi ∈ Bm holds for every i ∈ j.By the properties of the set An, there is a set b ⊂ j such that |b| > αnj and theconditions pi for i ∈ b have a lower bound p. By the properties of the set Bmapplied in the generic extension, it is possible to strengthen p and identify a setc ⊂ b such that |c| > βm|b| in such a way that p forces the set {qi : i ∈ c} tohave a lower bound in Q. Then the conditions 〈pi, qi〉 for i ∈ c have a commonlower bound in P ∗ Q. As the set c ⊂ j has size > αnβm|j|, this verifies thedesired properties of the set Cnm.

Claim 11.3.17. Let 〈Pn : n ∈ ω, πnm, ξmn : m ≤ n ∈ ω〉 be a very Suslin systemconsisting of Suslin measured forcings. Then the limit is Suslin measured.

Proof. Let Pω be the limit of the system as described in Definition 11.2.12. Foreach n ∈ ω, let Anm, εnm for m ∈ ω be analytic subsets of Pn and positiverational numbers witnessing the measured property of Pn. For each n,m ∈ ωlet Bnm ⊂ Pω be the set {〈p, n〉 ∈ Pω : p ∈ Anm}. It is immediate that the sets


Bnm with the numbers εnm witness the Suslin measured property of the limitPω.

A routine transfinite induction argument now completes the proof.

Theorem 11.3.18. The finite support product of countably many very Suslin,Suslin measured posets, if very Suslin, is Suslin measured.

Proof. Let Pn for n ∈ ω be the very Suslin posets, with their Suslin measuredproperty witnessed by sets Anm ⊂ Pn and numbers εnm for m ∈ ω. Considerthe finite support product Q =

∏n Pn and assume it is very Suslin. For each

function h ∈ ω<ω let Bh = {q ∈ Q : dom(q) = dom(h) ∧ ∀m ∈ dom(q) q(m) ∈Am,h(m), and let εh =

∏m∈dom(h) εm,h(m)}. It is not difficult to verify that the

sets Bh and the numbers εh witness the Suslin measured property of the posetQ.

As the last issue in this section, we elaborate on a notion which has been usedin several contexts to guarantee that various posets do not add dominating reals[70, 1]; we will use it for a different purpose.

Definition 11.3.19. Let P be a Suslin poset.

1. A set A ⊂ P is liminf-centered if for every collection {pn : n ∈ ω} ⊂ Athere is a condition q ∈ P such that every condition stronger than q iscompatible with pn for infinitely many numbers n ∈ ω;

2. P is Suslin σ-liminf centered if P =⋃nAn where each set An ⊂ P is

analytic and liminf-centered.

It is not difficult to see that the property of the condition q demanded by thefirst item of the definition can be stated in the forcing language as q thereare infinitely many n ∈ ω such that pn belongs to the generic filter. It isimportant and instructive to note that if P is a very Suslin c.c.c. poset then itsSuslin σ-liminf-centeredness will persist to all generic extensions. To see this,it is enough to observe that for an analytic set A ⊂ P , the statement “A isliminf-centered” is Π1

2; then, an application of Shoenfield absoluteness gives therequired conclusion. Now, the statement “A is liminf-centered” is equivalent tothe following: for all 〈pn : n ∈ ω〉, either for some n ∈ ω, pn /∈ A, or there isq ∈ P such that q there are infinitely many n ∈ ω such that pn belongs to thegeneric filter. The forcing statement is analytic by Proposition 11.2.9 and thevery Suslin assumption on P . Thus, the total statement is Π1

2 as required.

Theorem 11.3.20. Let P be a very Suslin c.c.c. forcing which is Suslin σ-liminf centered. Let α ∈ ω1 be a countable ordinal. Then the finite support of Pof length α is a very Suslin c.c.c. Suslin σ-liminf centered forcing.

Proof. The argument is a routine variation of the proofs of previous preservationtheorems.


Claim 11.3.21. Let P,Q be very Suslin c.c.c. forcings, both of which are Suslinσ-liminf-centered. The P ∗ Q is Suslin σ-liminf-centered.

Proof. Find covers Q =⋃mBm and P =

⋃nAn by analytic liminf-centered

sets. Let Cnm ⊂ P ∗ Q be the set of all conditions 〈p, q〉 such that thereexists a condition 〈p′, q′〉 ≤ 〈p, q〉 such that p′ ∈ An and p′ q′ ∈ Bm. It isnot difficult to check that P ∗ Q =

⋃nm Cnm and the sets Cnm are analytic

by Proposition 11.2.9. To check the liminf-centered property, suppose that{〈pi, qi〉 : i ∈ ω} are conditions in Cnm. Strengthening if necessary, we mayassume that pi ∈ An and pi qi ∈ Bm. Use the liminf-centeredness of An tofind a condition p forcing that for infinitely many i ∈ ω, pi is in the genericfilter. Using the liminf-centeredness of the set Bm, find a name q for a conditionin Q such that p q for infinitely many i ∈ ω such that pi belongs to thegeneric filter on P , qi belongs to the generic filter on Q. The condition 〈p, q〉 isas required.

Claim 11.3.22. Let 〈Pn : n ∈ ω, πnm, ξmn : m ≤ n ∈ ω〉 be a very Suslinsystem consisting of Suslin σ-liminf-centered forcings. Then the limit is Suslinσ-liminf-centered.

Proof. Let Pω be the limit of the system as described in Definition 11.2.12.Let Pn =

⋃mAnm : n ∈ ω be a cover by analytic liminf-centered sets for each

n ∈ ω. For each n,m ∈ ω let Bnm ⊂ Pω be the set {〈p, n〉 ∈ Pω : p ∈ Anm}.It is immediate that these are analytic liminf-centered analytic sets exhaustingPω.

The theorem follows by a routine transfinite induction argument.

The following question probably has a negative answer; still, products of thevery Suslin σ-liminf-centered posets used later in the chapter are again Suslinσ-liminf-centered.

Question 11.3.23. Is the class of Suslin σ-liminf-centered posets closed underproduct?

11.4 Locally countable simplicial complexes

In this section, we concentrate on tasks of a very specific form common to manyconcerns of modern descriptive set theory. These tasks start with a countableBorel equivalence relation E on a Polish space X and attempt to assign a struc-ture of some type to each E-class; at the same time, distinct E-classes do notcommunicate with each other in any way. Reviewing the tasks of this type, itbecomes obvious that often there is a locally countable simplicial complex Kon X such that the associated equivalence relation EK (as identified in Defini-tion 6.1.3) is E, and we seek a maximal K-set. Sometimes, the domain of thesimplicial complex K is instead the collection of finite subsets of X consistingof pairwise E-related points, and sometimes we actually seek a maximal K-set

11.4. LOCALLY COUNTABLE SIMPLICIAL COMPLEXES 271

with some additional largeness properties. The variations considered in thissection all fall under the following definition:

Definition 11.4.1. A locally countable pair is a pair 〈K,L〉 such that

1. K is a Borel locally countable simplicial complex on a Polish space X;

2. L ⊂ K is a cofinal Borel subset, i.e. ∀a ∈ K ∃b ∈ L a ⊂ b;

3. for every finite set a ⊂ X, a ∈ L just in case a ∩ c ∈ L holds for everyEK-class c ⊂ X.

A K-set a ⊂ X is L-regular if every finite subset of a is a subset of a finitesubset of a which is in L.

The role of the set L deserves a comment. Note that item (3) holds with Kreplacing L by the definition of the equivalence relation K; thus, distinct EK-classes do not communicate regarding the membership of finite sets in K and L.In many cases, the equality L = K will occur; some tasks seem to be impossibleto achieve in this way though. For example, let G be a graph on ω and let K bethe simplicial complex of all finite G-colorings with colors coming from ω. Notevery maximal K-set needs to be a total G-coloring. To alleviate this unwantedeffect, let L ⊂ K be the set of those finite colorings whose domain is a naturalnumber; then every L-regular maximal K-set is in fact a total G-coloring.

Definition 11.4.2. Let 〈K,L〉 be a locally countable pair on a Polish spaceX. The poset PKL is the poset of all sets p ⊂ K for which there is a countableEK-invariant set dom(p) ⊂ X such that p is a L-regular, maximal K-subset ofdom(p). The ordering is that of reverse inclusion. If K = L, we refer to theposet as PK instead.

It is immediate that the poset PKL is Suslin and σ-closed; in fact, it is a specialcase of the uniformization posets of Definition 6.6.5 with the space X × Xω,the equivalence relations E on X and F2 on Xω, and the invariant Borel setB ⊂ X × Xω consisting of all pairs 〈x, y〉 where y enumerates an L-regularmaximal K-subset of [x]E . The classification of balanced conditions then followsfrom Theorem 6.6.6:

Theorem 11.4.3. Let 〈K,L〉 be a locally countable pair on a Polish space X,with the associated poset PKL.

1. For every L-regular maximal K-set A ⊂ X, the pair 〈Coll(ω,X), A〉 isbalanced;

2. for every balanced pair 〈Q, σ〉 there is a set A as in (1) such that thebalanced pairs 〈Q, σ〉 and 〈Coll(ω,X), A〉 are equivalent;

3. distinct sets as in (1) yield inequivalent balanced pairs.


In order to identify properties of the poset PKL in the Solovay model, we haveto analyze its very Suslin c.c.c. control poset which in this case is simply thepartial order K ordered by reverse inclusion. Note that the set L does not enterthe definition of the control poset at all. The following proposition verifies thebasic property of the control poset.

Proposition 11.4.4. Let K be a Borel locally countable simplicial complex ona Polish space X. Then K is a very Suslin c.c.c. forcing.

Proof. Write E = EK; this is a Borel countable equivalence relation on thespace X. Observe that for an E-invariant set C ⊂ X, K ∩ P(C) is a regularsubposet of K. It is certainly closed under unions in K. Now suppose thata ∈ K is any condition and let b = a ∩ C. Whenever c ⊃ b is some condition inK ∩ P(C), a ∪ c ∈ K holds. To see this, recall that a ∪ c belongs to K just incase its intersection with every E-class is. Now, if d is an E-class, if d ⊂ C then(a ∪ c) ∩ d = c ∈ K, and if d ∩C = 0 then (a ∪ c) ∩ d = a ∈ K holds again. Theregularity of K ∩ P(C) in K has just been proved.

To verify the c.c.c. part of the proposition, let A ⊂ K is a maximal antichainand let M be a countable elementary submodel of a large structure containingA,E, and K. By the elementarity of M , the set A ∩M is a maximal antichainin the poset K ∩M = K ∩ P(X ∩M). Since E is an equivalence relation withall classes countable, the elementarity of M shows that X ∩M is an E-invariantset. By the previous paragraph then, A∩M is a maximal antichain in K. As aresult, A ∩M = A and so A is countable.

To verify the very Suslin part of the proposition, let A ⊂ K be a countableset. To check that A is a maximal antichain in K, it is only necessary tocheck that it is a maximal antichain in the poset K ∩ P(C), where C is theE-saturation of

⋃A by the first paragraph of the proof. Such a verification is

a Borel procedure as all E-classes are countable.

The key point in all subsections below will be the identification of some Suslinforcing preservation properties of the control poset and their connection to theforcing properties of the poset PKL in the Solovay model.

11.4a Suslin σ-centered complexes

The results of this section show that in certain extensions of the Solovay model,uncountable chromatic numbers of analytic hypergraphs of finite arity do notchange at all. This is the content of the following theorem.

Theorem 11.4.5. Let 〈K,L〉 be a locally countable pair. Suppose that thesimplicial complex K is Suslin σ-centered. Suppose that G is a analytic finitaryhypergraph of uncountable Borel chromatic number. Then in the PKL-extensionof the Solovay model, the chromatic number of G is uncountable.

Proof. As a preliminary consideration, it is clear from Fact 11.1.8 that it isenough to consider the case of G which is a principal skew product as in Defi-nition 11.1.7. Let 〈an, Hn, tn : n ∈ ω〉 be a sequence such that Hn = {an} andG =

∏nHn, tn.


Write X for the Polish space which is the domain of the Borel simplicialcomplex K, write E = EK and P = PKL. Let κ be an inaccessible cardinal, letW be the Solovay model derived from κ and work in the model W . Supposethat p ∈ P is a condition and τ is a P -name for a function from

∏n an to ω.

We must find a stronger condition p ≤ p, a number m ∈ ω and a G-hyperedgee such that p τ(y) = m for all vertices y ∈ e.

The condition p ∈ P as well as the name τ are definable from some parame-ters in the ground model and a parameter z ∈ 2ω. Let V [K] be an intermediatemodel obtained by a poset of size < κ containing the parameter z, and workin the model V [K] until further notice. Consider the iteration Q ∗ R where Qis the Cohen poset on

∏n an adding a generic point ygen ∈

∏n an, and in the

Q-extension, R is the finite support iteration of length ω1 of the poset K. Asusual, the poset K is reinterpreted at each stage of the iteration; we denotethe model obtained after α-th stage by Mα, so M0 is the Q-extension of V [K].Thus, the R poset adds a sequence of K-sets, 〈Aα : α ∈ ω1〉 derived from the it-eration components of R. By a density argument, for each E-class c representedin the model Vα, the set Aα ∩ c is a maximal K-subset of c which is L-regular.Consider the Q ∗ R-name p for the K-set defined in the following way: if c is anE-class represented in dom(p), then c ∩ p = c ∩ p, if c is an E-class representedin M0 but not in dom(p) then c ∩ p = A0 ∩ c, and if α ∈ ω1 is an arbitrarynonzero ordinal and c is an E-class represented in Mα but not in

⋃β∈αMβ then

c ∩ p = c ∩ Aα. It is not difficult to see that p is a Q ∗ R-name for a maximalK-set which is L-large, and therefore a balanced condition for the poset P byTheorem 11.4.3. Note that the condition does not depend on the point ygen butonly on the model V [K][ygen ].

Now, by a standard balance argument, Q ∗ R ∃m Coll(ω,< κ) p Pτ(ygen) = m. Let 〈q, r〉 be a condition in the iteration Q ∗ R deciding the valueof m; we abuse the notation and call the specific value m again. Strengtheningthe condition q ∈ Q if necessary, we may assume that

• there is an ordinal α ∈ ω1 such that q Q r is a name for a condition in

the initial segment Rα of the iteration R;

• by the iteration theorem 11.3.7, Rα is a very Suslin forcing which is Suslinσ-centered. Let {Bk : k ∈ ω} be analytic centered sets covering Rα. Wemay find a specific number k ∈ ω such that q r ∈ Bk;

• q = tj for some number j ∈ ω.

Let y ∈∏n an be a Cohen-generic element over the model V [K] extending tj

and move into the model V [K][y]. For each i ∈ aj let yi ∈∏n an be the element

obtained from y by rewriting the j-th entry with i. Thus, each yi is a Cohengeneric element over V [K] meeting the condition q, V [K][yi] = V [K][y], and{yi : i ∈ n} is a G-hyperedge. Let ri = r/yi. Thus, each ri is a condition inthe poset Rα, and even in the analytic centered set Bk. Thus, the conditionsri for i ∈ an have a common lower bound. Now, let H ⊂ R be a filter generic


over the model V [K][y] and containing all conditions ri for i ∈ an. Consider thebalanced condition p ≤ p in the poset P in the model V [K][y][H] described inthe third paragraph of the present proof. The forcing theorem applied for eachi ∈ an shows that for each i ∈ an, V [K][y][H] |= Coll(ω,< κ) p P τ(yi) = m.Thus, W |= p τ(yi) = m for each i ∈ an, completing the proof.

The verification of Suslin σ-centeredness for a given locally countable Borelsimplicial complex K may look like a formidable challenge. However, there is astandard trick, recorded in the following theorem, which makes short work of itin all cases we are aware of.

Theorem 11.4.6. Let K be a locally countable Borel simplicial complex ona Polish space X and write E = EK. Suppose that there is a Borel supportfunction supp: K → [X]<ℵ0 such that

1. supp(a) ⊂ [a]E;

2. whenever {ai : i ∈ j} are sets in K, subsets of the same E-class withpairwise disjoint supports, then

⋃i ai ∈ K.

Then K is Borel σ-centered.

Proof. Let Γ be a countable group acting on X in a Borel way such that Eis the resulting orbit equivalence relation. Let a ∈ K and write supp∗(a) =⋃{supp(a ∩ c) : c is an E-class}. We say that a tuple 〈b, f〉 is a descriptor of a

if

• b ⊂ B is a finite set consisting of pairwise disjoint basic open subsets ofX such that every element of supp∗(a) belongs to

⋃b and no element of

b contains more than one element of supp∗(a);

• f : b→ [Γ]<ℵ0 is a function such that whenever O ∈ b is an open set, c isan E-class, and x ∈ O∩supp(a∩c) is a point then a∩c = {γ ·x : γ ∈ f(O)}.

Note that a indeed has a descriptor since the set supp∗(a) is finite (making theconstruction of b as in the first item possible), and for any E-class c the setsupp(a∩c) is a subset of c by (1) (making the construction of f as in the seconditem possible). We will show that if {ai : i ∈ j} are elements of K with the samedescriptor 〈b, f〉 then

⋃i ai ∈ K. Since there are only countably many possible

descriptors, and the set of elements of K with a given descriptor is Borel, thiswill conclude the proof of the theorem.

Let c be an E-class; we must show that⋃i(ai ∩ c) ∈ K. In view of property

(2) of the supp function, it will be enough to show that whenever i0 6= i1 areelements of j then either ai0 ∩ c = ai1 ∩ c or supp(ai0 ∩ c) is disjoint fromsupp(ai1 ∩ c). Towards this end, suppose that x ∈ supp(ai0 ∩ c) ∩ supp(ai1 ∩ c)is a point; we have to show that ai0 ∩ c = ai1 ∩ c holds. Let O ∈ b be an openset such that x ∈ O, and use the second property of a descriptor to see thatai0 ∩ c = ai1 ∩ c = {γ · x : γ ∈ f(O)}. The theorem follows.


Our first example is motivated by a ZF+DC result of [19]: for every Borel locallyfinite graph on a Polish space X, if the chromatic number of G is ≤ n, thenthere is a meager set B ⊂ X and a decomposition of the set X \B into 2n− 1many Borel G-anticliques. We have:

Example 11.4.7. Let G be a Borel locally finite graph on a Polish space Xof chromatic number n. Let K be the simplicial complex of all finite partial Gcolorings with ≤ 2n − 1 colors which can be extended to a total coloring with≤ 2n − 1 many colors in which at most n many colors are attained infinitelymany times. Then K is a Borel locally countable simplicial complex which isSuslin σ-centered.

Proof. Note that K is in fact a simplicial complex on X× (2n− 1). The variousassertions of the example follow from two simple general graph theoretic claims.Let H be a locally finite graph on a set V of vertices and let c : V → m bea H-coloring with finitely many colors. Define F (c) = {i ∈ m : the set {v ∈V : c(v) = i} is finite}.

Claim 11.4.8. For every finite set a ⊂ V and every set e ⊂ m of size |F (c)|there is a H-coloring d : V → m such that c � a = d � a and F (d) = e.

Proof. It will be enough, given i ∈ F (c) and j ∈ m \ F (c), to find a coloringd : V → m such that c � a = d � a and F (d) = (F (c) ∪ {j}) \ {i}. To this end,let b = a ∪ {v ∈ V : c(v) = i} ∪ {u ∈ V : ∃v {u, v} ∈ H and c(v) = i}. Let dbe the coloring equal to c except on the vertices v ∈ V \ b such that c(v) = j,which will have d(v) = i. It is easy to verify that d works.

Claim 11.4.9. For every finite set a ⊂ V there is a finite partial H-coloringd : V → m such that a ⊂ dom(d), c � a = d � a, and for every edge {u, v} ∈ Hwith u ∈ dom(d) and v /∈ dom(d), d(u) /∈ |F (c)|+ 1 holds.

Proof. By the previous claim, we can adjust c so that F (c) is the set of thefirst F (c)-many natural numbers. Let a′ be the union of the finite sets a,{v ∈ V : c(v) ∈ F (c)}, and {v ∈ V : ∃u {u, v} ∈ H and c(u) ∈ F (c)}. Let b bethe union of the finite sets a′ and {v ∈ V : ∃u ∈ a′ {u, v} ∈ H and c(u) = |F (c)|}.Let d = c � b and observe that d works.

Now, we are ready to complete the proof of the example. Write E for the Borelequivalence relation of G-path connectedness on the space X. First of all, thesimplicial complex K is Borel and locally countable. Applying the last claimto each connected component of the graph G, it is clear that a finite partialG-coloring a belongs to K just in case for every E-class c ⊂ X there is a finitepartial G-coloring d such that dom(a) ∩ c ⊂ dom(d) ⊂ c and for every G-edge{x, y} with x ∈ dom(d) and y ∈ c \ dom(d) it is the case that d(u) /∈ n holds.(Such a coloring d can be completed by coloring the rest of the class c with thefirst n many colors, which is possible by the chromatic number assumption onG.) This is a Borel condition.


To prove the Suslin σ-centeredness of the simplicial complex K, use theLusin–Novikov theorem to find a Borel function g which, for each finite coloringa ∈ K whose domain consists of elements of a single E-class c, assigns a G-coloring g(a) ⊃ a such that dom(a) ⊂ dom(g(a)) ⊂ c and for every G-edge{x, y} with x ∈ dom(g(a)) and y ∈ c \ dom(g(a)) it is the case that g(a)(u) /∈ nholds. Let supp(a) be the set of all pairs 〈e, i〉 such that e ∈ dom(g(a)) andi ∈ 2n − 1. In view of Theorem 11.4.6, it will be enough to show that if{ai : i ∈ j} are colorings in K whose domain is a subset of one and the same E-class c, with pairwise disjoint supports, then

⋃i ai ∈ K. This is nearly trivial,

however. The colorings g(ai) for i ∈ j have pairwise disjoint domains. Letf : c \

⋃i dom(g(ai)) → n be any G-coloring–its existence is guaranteed by the

chromatic number assumption on G. Then, f ∪⋃i g(ai) is a G-coloring of

the class c which extends⋃i ai and uses no colors ≥ n infinitely many times,

witnessing that⋃i ai ∈ K.

Corollary 11.4.10. Let G be a locally finite Borel graph on a Polish space X offinite chromatic number n. Let K be the simplicial complex as in Example 11.4.7.Let H be a finitary analytic hypergraph of uncountable Borel chromatic number.Then

1. in the PK-extension of the Solovay model, the chromatic number of H isuncountable;

2. it is consistent relative to an inaccessible cardinal that ZF+DC holds andthe chromatic number of G is ≤ 2n− 1 and the chromatic number of H isuncountable.

The following example is motivated by a question of Marks and Unger (privatecommunication): given a countable Borel equivalence relation E, is it consis-tent with ZF+DC that E is a union of an increasing countable sequence ofequivalence relations with all equivalence classes finite and every set of realshas the Baire property? While that question remains open, we do have partialinformation:

Example 11.4.11. Let E be a countable Borel equivalence relation on a Polishspace X with all classes infinite. Let K be the simplicial complex of all finitedirected subgraphs of E which are acyclic and every vertex has in-degree ≤ 1and out-degree ≤ 1. Note that K is in fact a simplicial complex on E ratherthan on X. The simplicial complex K is Suslin σ-centered by Theorem 11.4.6.To see this, let Γ be a countable group acting on X in a Borel way such that Eis the resulting orbit equivalence relation. Let Γ = {γk : k ∈ ω} be an arbitraryenumeration. Whenever a ∈ K is a graph whose domain is a subset of a single E-class, let n(a) be the smallest number such that for any two points x, y ∈ dom(a)there is k ∈ n(a) such that x = γk · y, and let supp(a) = {{γl · x, γk · x} : x ∈dom(a), k, l ∈ n(a)}. If two graphs a0, a1 ∈ K have disjoint support, they musthave disjoint domains. It follows that if ai ∈ K are sets for i ∈ j which aresubsets of the same E-class and have pairwise disjoint supports, then

⋃i ai ∈ K

as required in Theorem 11.4.6.


We plan to use the simplicial complexK to introduce an ordering of ordertypeZ to each E-class. However, not every maximal K-set does that. Apparently, acofinal subset L ⊂ K is needed so that every L-large maximal K-set comes closethe the purpose in mind. For one simple solution, let L ⊂ K be the collectionof all sets a ∈ K which in each E-class have only one vertex of in-degree 0 andonly one vertex of out-degree 0. Note that L ⊂ K is cofinal, and if c ⊂ X is anE-class and d ⊂ c2 is an L-regular maximal K-set, then d is a ray of ordertypeZ or N or inverted N.

Corollary 11.4.12. Let E be a countable Borel equivalence relation with allclasses infinite as in Example 11.4.11. Let H be a finitary analytic hypergraphof uncountable Borel chromatic number.

1. In the PKL-extension of the Solovay model, the chromatic number of H isuncountable;

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds andE is the orbit equivalence relation of a (discontinuous) Z-action and thechromatic number of H is uncountable.

Proof. The first item is the consequence of the fact that K is Suslin σ-centeredas proved in Example 11.4.11. For the second item, let W be the Solovay modelderived from some inaccessible cardinal and let G ⊂ PKL be a generic filter overthe model W . A genericity argument shows that

⋃G is an L-regular maximal

K-set. For every E-class c,⋃G on the class c indicates either an ordering of

type Z or of type N or of type inverted N. It is a matter of minor surgery inZF+DC to turn this into a relation which is an ordertype Z on every class.Thisimmediately provides an action of Z on X which yields E as an orbit equivalencerelation. The chromatic number part of the second item follows from the firstitem.

As a final observation, we show that the forcings of the type discussed inTheorem 11.4.5 can be placed in countable support product, and the conclusionof Theorem 11.4.5 remains in place. Suppose that 〈Xi : i ∈ ω〉 are pairwisedisjoint Polish spaces, and 〈Ki,Li〉 are locally countable pairs on each. Supposethat each simplicial complex Ki is Suslin σ-centered. Let X be the Polish spacewhich is the disjoint union of Xi for i ∈ n, let K be the simplicial complex ofthose finite sets a ⊂ X such that a ∩ Xi ∈ Ki holds for every i ∈ ω, and letL = {a ∈ K : ∀i a ∩Xi 6= 0 → a ∩Xi ∈ Li. Then PKL is a countable supportproduct of the posets PKiLi , and the simplicial complex K is Suslin σ-centeredby Theorem 11.3.8, since it is the finite support product of the posets Ki fori ∈ ω. Therefore, by Theorem 11.4.5, in the

∏i PKiLi-extension of the Solovay

model, the uncountable chromatic numbers of finitary analytic hypergraphs arepreserved.


11.4b Suslin σ-linked complexes

In fairly common situations, one wishes to preserve uncountable chromatic num-bers of analytic graphs while a task of higher arity is performed. This can behandled by Suslin σ-linked complexes as explained in the following theorem.

Theorem 11.4.13. Let 〈K,L〉 be a locally countable pair. Suppose that thesimplicial complex K is Suslin σ-linked. Let G be an analytic graph with un-countable Borel chromatic number. In the PKL-extension of the Solovay model,the chromatic number of G is uncountable.

The verification of the Suslin σ-linked property of Borel simplicial complexes isinvariably achieved via the following result.

Theorem 11.4.14. Let K be a locally countable Borel simplicial complex ona Polish space X and write E = EK. Suppose that there is a Borel supportfunction supp: K → [X]<ℵ0 such that

1. supp(a) ⊂ [a]E;

2. whenever a0, a1 are sets in K, subsets of the same E-class with disjointsupports, then a0 ∪ a1 ∈ K.

Then K is Borel σ-linked.

The proofs of Theorems 11.4.13 and 11.4.14 are nearly identical to the proofsof Theorem 11.4.5 and 11.4.6, and we omit them.

Our first example is motivated by a result of Marks and Unger [68] whichfound perfect matchings with the Baire property in many Borel bipartite graphs.In order to do this, they isolated the following strengthening of the well-knownHall’s condition of [38]:

Definition 11.4.15. Let Γ be a locally finite bipartite Borel graph on a Polishspace X. We say that Γ satisfies the Marks–Unger condition if there is a realnumber ε > 0 such that for every finite set a ⊂ X on one side of the bipartition,the number of Γ-neighbors of a is at least (1 + ε)|a|.

Example 11.4.16. Let G be a locally finite bipartite Borel graph on a Polishspace X satisfying the Marks–Unger condition. Let K be the simplicial complexon the edges of G consisting of all sets b ⊂ G which can be completed to a perfectmatching of G. Then K is Suslin σ-linked.

Proof. Let ε > 0 be a real number witnessing the Marks–Unger property of thegraph G. Let E = EK be the G-path equivalence relation on X. Let a ∈ K be aset which is a subset of a single E-class; we must calculate the support supp(a)such that the assumptions of Theorem 11.4.14 are satisfied.

Let supp(a) be the set of all G-edges which are within G-distance at most8|a|/ε from some edge in a. We must verify the amalgamation condition, i.e. ifa, b ∈ K are disjoint finite sets in the same E-class such that supp(a)∩supp(b) =0, then a∪b ∈ K. By Hall’s marriage theorem [38], it is enough to verify that the


graph G′ obtained from G by erasing all vertices mentioned in a or b satisfies theHall’s marriage condition: for each finite set c of vertices in the same equivalenceclass as a, b on one side of the bipartition, the set of G′-neighbors of c has sizeat least |c|. There are two cases:Case 1. There is a G2-path from some vertex in a to some vertex in b usingonly nodes in c as the intermediate steps. In this case, without loss of generalitysuppose that |a| ≥ |b|, and use the definition of supp(a) to argue that the G2-path from a to b must contain at least 4|a|/ε-many steps. This means that|c| > 4|a|/ε. By the initial condition on the graph G, the set c has at least(1 + ε)|c| many G-neighbors and so at least (1 + ε)|c| − 2|a| − 2|b| many G′-neighbors. However, 2|a| + 2|b| < ε|c| and so the set c has at least |c|-manyG′-neighbors as required.Case 2. Case 1 fails. In this case, the set c can be written as a union of disjointsets c = ca ∪ cb such that the set ca has no common G-neighbors with cb, thesets ca and

⋃b are G-disconnected, and the sets cb and

⋃a are G-disconnected.

Then, since a ∈ K, the set ca has at least |ca|-many G′-neighbors; since b ∈ K,the set cb has at least |cb|-many G′-neighbors. The G′-neighborhoods of ca, cbdo not overlap by the choice of ca and cb, and so the set c has at least |c|-manyG′-neighbors as desired.

Note that as the graph G is locally finite, maximal K-sets are in fact perfectG-matchings.

Corollary 11.4.17. Let G be a locally finite Borel bipartite graph on a Polishspace X satisfying the Marks–Unger condition. Let K be the simplicial complexof partial G-matchings that can be extended to a perfect matching. Let H be ananalytic graph of uncountable Borel chromatic number.

1. In the PK-extension of the Solovay model, the chromatic number of H isuncountable;

2. it is consistent relative to an inaccessible cardinal that ZF+DC holds, thereis a perfect G-matching, and the chromatic number of H is uncountable.

Question 11.4.18. Is there a locally finite, bipartite Borel graph G satisfyingthe Marks–Unger condition such that the associated control poset is not Suslinσ-centered? Such that in ZF+DC, the existence of perfect matching for G isequivalent to some Borel hypergraph having countable chromatic number?

To many hypergraphs of arity greater than two it is possible to add a coloringwhile preserving uncountable chromatic numbers of analytic graphs purely as amatter of arity. Consider the following:

Example 11.4.19. Let Z act on a Polish space X in a Borel free way. Let Gbe the hypergraph of all triples {x0, x1, x2} such that there is n ∈ ω such thatn · x0 = x1 and n · x1 = x2. Let K be the simplicial complex of partial finitecolorings of the hypergraph G with colors coming from ω. The complex K isSuslin σ-linked.


Proof. Note that K is in fact a simplicial complex on X×ω. It is easy to observethat E = EK is the equivalence relation on X × ω connecting 〈x, n〉 and 〈y,m〉just in case x, y ∈ X are orbit-equivalent. Let a ∈ K be a nonempty finitepartial G-coloring in a single E-class. We will compute a support supp(a) suchthat if a, b ∈ K are subsets of the same E-class and supp(a) ∩ supp(b) = 0 thena ∪ b ∈ K. This will be enough by Theorem 11.4.14.

Let n(a) ∈ ω be the smallest number larger than all elements of rng(a) andall numbers n such that there are points x0, x1 ∈ dom(a) with n · x0 = x1.Let supp(a) = {〈m · x, k〉 : x ∈ dom(a), k ∈ n(a) and m ∈ Z is of absolutevalue smaller than n(a)}. We claim that the function supp works as required inTheorem 11.4.14. Suppose that a, b ∈ K are colorings in the same E-class withdisjoint supports; we must show that a ∪ b ∈ K. It is not hard to check thata ∪ b is in fact a function. To see that it is a G-coloring, let {x0, x1, x2} ∈ Gbe a hyperedge which is a subset of dom(a) ∪ dom(b); we must show that thecoloring a∪b is not constant on it. If it is a subset of dom(a) or dom(b), then weare done as both a, b are G-colorings. Otherwise, two of the points (say x0, x1)would have to be in the domain of one of the colorings (say a) and the remainingpoint (z) has to be in the domain of the other coloring (b). If b(z) 6= a(x0) thenwe are done. However, b(z) = a(x0) is impossible: then b(z) ∈ n(a) and so〈z, b(z)〉 ∈ supp(a) ∩ supp(b), violating the initial assumptions on a, b.

In order to force a total G-coloring, we need to choose a cofinal subset L ⊂ Ksuch that L-regular maximal K-sets are in fact total colorings. One possiblechoice is L = {a ∈ K : for every infinite E-class c ⊂ X, dom(a) ∩ c is asubinterval of the shift order on c, and the values of a at the endpoints ofthis interval are larger than the length of the interval}. The verification of therequisite property of L is easy and left to the reader.

Corollary 11.4.20. Let G be the hypergraph and 〈K,L〉 be the locally countablepair of Example 11.4.19. Let H be an analytic graph of uncountable chromaticnumber.

1. In the PKL-extension of the Solovay model, the chromatic number of H isuncountable;

2. it is consistent relative to an inaccessible cardinal that ZF+DC holds andthe chromatic number of G is countable while the chromatic number of His uncountable.

11.5 Larger graphs

In this section, we study the options for adding Γ-colorings with countablymany colors to various, not necessarily locally countable Borel graphs Γ onPolish spaces while preserving the main features of locally countable structures.The main tool is the control forcing of finite approximations to such colorings,as recorded in the following definition.

11.5. LARGER GRAPHS 281

Definition 11.5.1. Let Γ be a Borel graph on a Polish space X. KΓ is thesimplicial complex on X×ω consisting of finite sets a ⊂ X×ω which are partialΓ-colorings. KΓ is viewed as a poset ordered by reverse inclusion.

In order to extract desirable properties of the poset KΓ, we deal with graphswith forbidden subgraphs. The following definition records the patterns we wishto prohibit later; all of them are bipartite graphs.

Definition 11.5.2. Let n ∈ ω be a natural number.

1. Kn,n is the bipartite graph on 2× n connecting any pair of vertices withdistinct first coordinates;

2. Kn,ω1is the bipartite graph on the disjoint union of n and ω1 connecting

all elements of n to all elements of ω1;

3. K→ω,ω is the graph on 2×ω consisting of all pairs {〈0,m〉, 〈1, k〉} such thatm ∈ k.

The first three theorems of this section deal with the definability and regularityproperties of the control forcing.

Theorem 11.5.3. Let Γ be a Borel graph on a Polish space X which for somen ∈ ω does not contain an injective homomorphic image of Kn,ω1

. Then KΓ isa very Suslin c.c.c. partial ordering.

Note that in this case the graph Γ has countable coloring number by a resultof [25]. Observe also that the hypothesis on the graph is absolute throughoutall forcing extensions. It simply says that for all injective tuple ~x ∈ Xn, the set{y ∈ X : ∀i ∈ n ~x(i) Γ y} is countable. By the perfect set theorem for Borelsets, this is equivalent to the statement that for all injective tuples ~x ∈ Xn andall continuous maps π : 2ω → X, either there are distinct points z0, z1 ∈ 2ω suchthat π(z0) = π(z1) or there is a point z ∈ 2ω and a number i ∈ n such thatπ(z) Γ ~x(i) fails. A closer reading of the formula reveals that it is ≯1

2 and sothe Shoenfield absoluteness applies to it.

Proof. Write K = KΓ. It is clear that K is a Suslin poset. For the c.c.c. partof the theorem, let A = {aα : α ∈ ω1} be an uncountable set; we must producetwo compatible elements in it. Let M be a countable elementary submodel ofa large structure containing Γ and A. Let α ∈ ω1 \M be any ordinal, and letb = aα � M . By the elementarity of the model M , there have to be infinitelymany ordinals β such that b ⊂ aα.

For each y ∈ dom(aα) \ b, the set uy = {x ∈ M ∩X : x Γ y} must have sizeless than n: otherwise, one can find distinct elements {xi : i ∈ n} ⊂ y and placey in the set {z ∈ X : ∀i ∈ n xi Γ z}. The latter set is countable by the initialassumption on Γ, it is an element of the model M and by elementarity of M itis a subset of M . This would contradict the choice of the point y.

Now, by elementarity of the model M , there is an ordinal β ∈M ∩ ω1 suchthat b ⊂ dom(aβ) and dom(aβ) ∩

⋃y uy = dom(b). It is not difficult to check

that aα, aβ are compatible as desired.


We turn to the proof of the very Suslin property. By the assumptions, theset B = {〈~x, y〉 ∈ Xn ×X : ~x is injective and ∀i ∈ n ~x(i) Γ y} is Borel and allof its vertical sections are countable by the initial assumption on the graph Γ.Use the Lusin–Novikov theorem to find Borel functions {fi : i ∈ ω} from Xn toX such that B ⊂

⋃i fi.

Now, let C ⊂ K be a countable set. Write supp(C) ⊂ X for the closureof the set

⋃c∈C dom(c) under the functions fi for i ∈ ω. We claim that the

following are equivalent:

• C is predense;

• for every b ∈ K with dom(b) ⊂ supp(C) there is c ∈ C such that c∪ b ∈ K.

This will show that the collection of (enumerations of) countable predense sub-sets of K which are predense is Borel, since the second item clearly gives a Boreldescription of it.

Now, the first item certainly implies the second. The opposite implicationis the heart of the matter. Suppose that the first item fails, and let a ∈ K bea condition such that for every c ∈ C c ∪ a /∈ K holds. Note that for everyy ∈ dom(a) \ supp(C), the set uy = {x ∈ supp(C) : x Γ y} must have size lessthan n, otherwise y ∈ supp(C) would hold by the initial choice of the functionsfi. Strengthening a if necessary then, we may assume that

⋃y uy ⊂ dom(a).

Let b = a � supp(C) and argue that b witnesses the failure of the second item.Let c ∈ C be an arbitrary element. Since c is incompatible with a, there

must be either some x ∈ dom(c) ∩ dom(a) such that c(x) 6= a(x), or some Γ-connected x ∈ dom(c) and y ∈ dom(a) such that c(x) = a(y). In the formercase, x ∈ supp(C) and so x ∈ dom(b) witnesses the incompatibility of b, c. Inthe latter case, if y ∈ supp(C) then y ∈ dom(b) witnesses the incompatibilityof b, c. If y /∈ supp(C) then x ∈ dom(b) by the strengthening of a, and thenb(x) 6= a(y) = c(x) and so x witnesses the incompatibility of b, c again.

Theorem 11.5.4. Let Γ be a Borel graph on a Polish space X such that thereis no injective homomorphism from K→ω,ω to Γ. Then KΓ ordered by reverseinclusion is a Suslin σ-liminf-centered poset.

Note that we are not asserting that the poset KΓ must be very Suslin in thiscase; we do not know that.

Proof. For each number n ∈ ω let An = {a ∈ K : |dom(a)| < n and rng(a) ⊂ n}.It is clear that each set An is Borel and K =

⋃nAn. Assume that one of the

sets An is not liminf-centered and work to produce an injective homomorphismfrom K→ω,ω to Γ. This will prove the theorem.

Suppose that {ai : i ∈ ω} is a collection of conditions in An such that it isoutright forced by K that ai belongs to the generic filter for only finitely manyvalues of i ∈ ω. Let b ∈ K be inclusion-maximal such that b ⊂ ai holds forinfinitely many i ∈ ω; thinning out the original collection if necessary, we mayassume that b ⊂ ai holds for all i ∈ ω. Thinning out even further, we mayassume that the sets dom(ai \ b) for i ∈ ω are pairwise disjoint.


By recursion on j ∈ ω build conditions cj ∈ K such that

• b ⊂ cj ;

• for all x ∈⋃k∈j dom(ck \ b) ∪

⋃k∈j dom(ak \ b), cj(x) > n;

• for all but finitely many numbers i ∈ ω, cj ∪ ai /∈ K.

This is possible by the initial assumptions on the collection {ai : i ∈ ω}. Notethat the first and third item imply that for a given j ∈ ω, for all but finitelymany numbers i ∈ ω there are elements x ∈ dom(cj) and y ∈ dom(ai \ b) suchthat x Γ y and cj(x) = ai(y) both hold. In addition, the point x ∈ X cannotbelong to

⋃k∈j dom(ck) ∪

⋃k∈j dom(ak) by the second item.

Now, let ≺ be any linear ordering of the space X. For numbers j, t writex(j, t) for the t-th element of dom(cj) \

⋃k∈j dom(ck)∪

⋃k∈j dom(ak) in the or-

dering ≺. Similarly, write y(i, s) for the s-th element of dom(ai) in the ordering≺. Let U be a nonprincipal ultrafilter on ω. For each j ∈ ω there are numberstj ∈ |dom(cj)| and sj ∈ n such that the set aj = {i ∈ ω : x(j, tj) Γ y(i, sj)}belongs to the ultrafilter U . Also, there is a set d ⊂ ω in U and a number s ∈ ωsuch that for all j ∈ d, sj = s holds. Now by recursion on k ∈ ω define anincreasing sequence of numbers jk so that jk ∈ d ∩

⋂l∈k ajl . Finally, define the

map π : 2× ω → X by π(0, k) = x(j2k, tj2k) and π(1, k) = y(j2k+1, s). It is notdifficult to see that π is an injective homomorphism of K→ω,ω to Γ.

Theorem 11.5.5. Let Γ be a Borel graph on a Polish space X containing noinjective homomorphic image of Kn,n for some number n ∈ ω. Then the posetKΓ is Suslin σ-Ramsey-centered.

Proof. Write K = KΓ. For a number m ∈ ω let Am = {c ∈ K : |dom(c)| ≤ mand rng(m) ⊂ m}. Clearly, the sets Am are all Borel and K =

⋃mAm. It will be

enough to prove that each number m ∈ ω, the set Am ⊂ K is Ramsey-centered.Let k ∈ ω be an arbitrary number greater than 2n. We have to find l ∈ ω

such that every set {ci : i ∈ l} ⊂ Am contains a subset of size k with a commonlower bound. We will show that any number l such that l → (k)2

2m2+1 works.To see this, let {ci : i ∈ l} ⊂ Am be a set. Let ≺ be any linear ordering ofthe space X, for each i, u ∈ ω write ciu for the u-th element of dom(ci) in theordering ≺ if it exists, and define a partition π : [l]2 → (m ×m × 2) ∪ {∞} byrequiring the following. If i ∈ j ∈ l and π(i, j) = 〈u, v, 0〉 then ciu = cjv andci(ciu) 6= cj(cjv); if π(i, j) = 〈u, v, 1 then ciu Γ cjv and ci(ciu) = cj(cjv); and ifπ(i, j) =∞ then no u, v as in the previous items can be found.

Use the Ramsey property of the number l to find a set a ⊂ l of size k whichis homogeneous for the partition π. It is enough to argue that the set {ci : i ∈a} has a common lower bound. To see this, inquire about the homogeneouspartition value achieved. It cannot be of the form 〈u, v, 0〉 because then thecoloring ci has irreconcilable candidates for the value of ci(ciu), where i is thesecond number in the set a. The homogeneous partition value cannot be ofthe form 〈u, v, 1〉 since then the points ciu where i ranges over the first n manyelements of a, and cjv where j ranges over the second n-many elements of a,


form an injective homomorphic copy of Kn,n in Γ, contradicting the initialassumptions on Γ. Thus, the homogeneous color is∞, which precisely says that⋃i∈a ci is a function which is a Γ-coloring and therefore a common lower bound

of the conditions ci for i ∈ a.

Finally, we are ready to prove some preservation results.

Theorem 11.5.6. Let Γ be a Borel hypergraph on a Polish space X contain-ing no injective homomorphic image of K→ω,ω and Kn,ω1 for some n ∈ ω. Let∆ be a Borel graph which does not have Borel σ-finite clique number. In thePΓ-extension of the symmetric Solovay model, the chromatic number of ∆ isuncountable.

Proof. By Fact 11.1.26 it is enough to prove this for the graph ∆ on ωω whichis the skew product of countably many cliques on ω. Write also P = PΓ andK = KΓ. Let κ be an inaccessible cardinal, let W be the symmetric Solovaymodel derived from κ, and work in W . Suppose that p ∈ P is a conditionand τ is a P -name such that p τ : ωω → ω is a function. We must find acondition p ≤ p, ∆-connected points y0, y1 ∈ ωω and a number m such thatp τ(y0) = τ(y1) = m.

Strengthening the condition p if necessary, we may assume that wheneverxi for i ∈ n are distinct elements of dom(p) and y ∈ X is a point Γ-connectedto them all, then y ∈ dom(p) holds. The condition p and the name τ are alldefinable in the model W from ground model parameters and som additionalparameter z ∈ 2ω. Let V [K] be an intermediate model obtained by forcing ofsize less than κ such that z ∈ V [K] and work in V [K]. Consider the iterationQ ∗ R where Q is the Cohen forcing Q on ωω, and R is (forced to be) the finitesupport iteration of length ω1 whose iterands are the posets K evaluated in theirrespective models. Let Mα denote the model obtained after the initial segmentof the iteration R of length α and let ηα be the union of the generic filter on Kat the α-th stage of the iteration. Thus, R forces ηα to be a Γ-coloring on thedomain X ∩Mα.

To smooth out the various colorings ηα to a single total Γ-coloring extendingp, we need a small claim:

Claim 11.5.7. Let y ∈ X ∩ Mα \⋃β∈αMβ be a point. Then the set {x ∈

X ∩⋃β∈αMβ : x Γ y} has size less than n.

Proof. If this failed, then (regardless of whether α is limit or successor) therewould be a fixed ordinal β ∈ α and pairwise distinct points xi ∈ X ∩Mβ fori ∈ n such that xi Γ y holds for all i ∈ n. But then, by the initial assumption onΓ the set of all points simultaneously connected to all xi for i ∈ n is countable,coded in Mβ , and so a subset of Mβ . This means that y ∈ Mβ , contradictingthe initial assumptions on the point y.

Let {bk : k ∈ ω} be a recursive partition of ω into infinite sets. Now, define theQ∗R-name σ for a total Γ-coloring by the following recursive formula. σ �M0 isdefined for a point y ∈M0 so that if y ∈ dom(p) then σ(y) = p(y); if y /∈ dom(p)


then σ(y) is the smallest element of bη0(y) which is distinct from all colors p(x)for points x ∈ dom(p) which are Γ-connected with y. If α > 0 then let σ(y) bethe smallest element of bηα(y) which is distinct from all colors p(x) for pointsx ∈ X ∩

⋃β∈αMβ which are Γ-connected with y. Note that the definition is

correct as the number of colors in the set bηα(y) we must avoid is at most n bythe claim and the initial assumption on p.

Now, observe that σ is really a name for a Γ-coloring. To see this, supposethat y0, y1 ∈ X are Γ-connected points which appear in the models Mα0

andMα1

respectively. For definiteness assume that α0 ≤ α1 holds. If α0 = α1,then σ(y0) 6= σ(y1) must hold as η(y0) 6= η(y1) holds. If α0 < α1 then σ(y1)is distinct from all colors of points assigned to the points Γ-connected to y1

and belonging to the models with smaller index, in particular σ(y1) 6= σ(y0).Observe also that the name σ does not depend on the Q-generic point, but onlyon the model this point generates over V [K].

Total Γ colorings are balanced virtual conditions in P by Theorem 8.1.2.Thus, writing χ for the Q-name for the generic element of ωω added by theposet Q, a balance argument shows that Q ∗ R Coll(ω,< κ) σ decides inP the value of τ(χ). Let 〈q, r〉 be a condition in the iteration Q ∗ R and letm ∈ ω be a number such that 〈q, r〉 Coll(ω,< κ) σ τ(χ) = m. By a chaincondition argument, there is a countable ordinal α such that q r ∈ Rα, theintial segment of the iteration R of length α.

The poset K is very Suslin and Suslin σ-liminf-centered by Theorems 11.5.3and 11.5.4. It follows by Theorem 11.3.20 that Rα is Suslin σ-liminf-centered; letRα =

⋃k Ak be its cover by analytic liminf-centered pieces. Strengthening q if

necessary, we may assume that there is k ∈ ω such that q r ∈ Ak; in addition,q is one of the finite sequences used in the definition in the skew product ∆.Let y ∈ ωω be a point Cohen generic over V [K] such that q ⊂ y, and for eachnumber l ∈ ω let yl ∈ ωω be the point resulting from rewriting the first entry ofy past q with l. Thus, the points yl form a ∆-clique, they are all Cohen-genericover V [K], and they all generate the model V [K][y]. By the liminf-centerednessof the set Ak, there is a condition s ∈ Rα which forces the set {l ∈ ω : r/ylbelongs to the generic filter} to be infinite. Let H ⊂ R be a filter generic overV [K][y] meeting the condition s. In V [K][y][H], let p = σ/H; this is a balancedcondition for the poset P . Let yl0 , yl1 ∈ ωω be two distict points such that theconditions r/yl0 , r/yl1 both belong to the filter H. The forcing theorem appliedin the model V [K] shows that in the model W , p P τ(yl0) = τ(yl1) = m asdesired.

Theorem 11.5.8. Let Γ be a Borel hypergraph on a Polish space X containingno injective homomorphic image of Kn,n for some n ∈ ω. Let ∆ be a Borelgraph which does not have Borel σ-bounded clique number. In the PΓ-extensionof the symmetric Solovay model, the chromatic number of ∆ is uncountable.

Proof. In view of Fact 11.1.24, it is enough to prove the theorem for a specificgraph ∆. Let Y be the product

∏n(n+ 1) and let ∆ be the graph on Y which

is the skew product of cliques on the sets n + 1 for n ∈ ω. The proof now


proceeds just as in Theorem 11.5.6; we only indicate the differences. The spaceωω is naturally replaced with Y . The last paragraph is then amended to thefollowing.

The poset K is very Suslin and Suslin σ-Ramsey-centered by Theorems 11.5.3and 11.5.5. It follows by Theorem ?? that Rα is Suslin σ-Ramsey-centered; letRα =

⋃k Ak be its cover by analytic Ramsey-centered pieces. Strengthening q if

necessary, we may assume that there is k ∈ ω such that q r ∈ Ak. Let m ∈ ωbe such that any m-tuple of elements of Ak contains two compatible conditions.Strengthening q further if necessary, we may assume that q is one of the finitesequences used in the definition in the skew product ∆, and the length of q isgreater than m. Let y ∈ Y be a point Cohen generic over V [K] such that q ⊂ y,and for each number l ∈ |q| let yl ∈ ωω be the point resulting from rewritingthe first entry of y past q with l. Thus, the points yl form a ∆-clique, theyare all Cohen-generic over V [K], and they all generate the model V [K][y]. Bythe choice of the number m, there are distinct numbers l0, l1 ∈ |q| such that theconditions r/yl0 and r/yl1 are compatible, with a lower bound denoted by s. LetH ⊂ R be a filter generic over V [K][y] meeting the condition s. In V [K][y][H],let p = σ/H; this is a balanced condition for the poset P .The forcing theoremapplied in the model V [K] shows that in the model W , p P τ(yl0) = τ(yl1) = mas desired.

Example 11.5.9. Let Γ be the graph H<ω. It does not contain an injectivehomomorphic copy of K2,ω, since for any point x, the points Γ-connected to itform a sequence converging to x; therefore, any two distinct points can haveonly finitely many common neighbors.

Theorem 11.5.6 now yields

Corollary 11.5.10. Let Γ be the graph H<ω.

1. In the PΓ-extension of the Solovay model, the graph Hω has uncountablechromatic number. In particular, there is no E0-transversal;

2. it is consistent relative to an inaccessible cardinal that ZF+DC holds andthe chromatic number of H<ω is countable and the chromatic number ofHω is uncountable.

Example 11.5.11. Let ~r be a sequence of positive real numbers converging tozero. Let Γ be the graph on X = R2 connecting two points if their Euclideandistance is on the sequence ~r. The graph Γ does not contain an injective ho-momorphic copy of K3,ω: if x0, x1 ∈ X are distinct points and ε > 0 is somereal number, then there are only finitely many points in the plane which areΓ-connected to both and of > ε-distance to both. Now, if x0, x1, x2 ∈ X aredistinct points and ε > 0 is smaller than half of the minimum distance betweentwo of the three, we see that every point of the plane is at a distance > ε fromtwo of the three, and so there are only finitely many points Γ-connected to allthree.

11.6. COLLAPSES 287

Theorem 11.5.6 now yields

Corollary 11.5.12. Let ~r be a sequence of positive real numbers converging tozero. Let Γ be the graph on X = R2 connecting two points if their Euclideandistance is on the sequence ~r.

1. In the PΓ-extension of the Solovay model, the graph Hω has uncountablechromatic number. In particular, there is no E0-transversal;

2. it is consistent relative to an inaccessible cardinal that ZF+DC holds, thechromatic number of Γ is countable and the chromatic number of Hω isuncountable.

Example 11.5.13. Let D ⊂ R be a Borel set algebraically independent overQ, consisting of positive reals. Let Γ be the graph on X = R2 connecting twopoints if their Euclidean distance belongs to D. The graph Γ does not containan injective homomorphic copy of Kn,n for some large number n [59, Theorem1]

Corollary 11.5.14. Let D ⊂ R be a Borel set algebraically independent overQ, consisting of positive reals. Let Γ be the graph on X = R2 connecting twopoints if their Euclidean distance belongs to D.

1. In the PΓ-extension of the Solovay model, the diagonal Hamming graphH<ω has uncountable chromatic number;

2. it is consistent relative to an inaccessible cardinal that ZF+DC holds, thechromatic number of Γ is countable and the chromatic number of H<ω isuncountable.

11.6 Collapses

In this section, we consider the problems of introducing an injection from |F | to2ω or to E0, for a given pinned Borel equivalence relation F . It turns out thatthe natural forcings preserve uncountable chromatic numbers of many graphsand hypergraphs. In particular, they do not introduce a total E0-transversal.We start with the collapse to 2ω.

Let E be a pinned equivalence relation on a Polish space X. Recall the defi-nition of the poset collapsing |E| to |2ω| from Definition 6.6.2: PE be the partialorder of pairs p = 〈ap, bp〉 where ap : X → 2ω is a countable partial functionsuch that for x0, x1 ∈ dom(ap), x0 E x1 iff ap(x0) = ap(x1), and bp ⊂ 2ω isa countable set disjoint from rng(ap). The ordering is that of coordinatewisereverse inclusion. It is immediate in ZF+DC that the ordering PE is Suslin andσ-closed, and the union of the PE-generic filter is an injection from the set ofE-classes to 2ω. The balanced conditions are classified by all such injections asproved in Theorem 6.6.3.


Theorem 11.6.1. Let E be a pinned Borel equivalence relation on a Polishspace X. Suppose that G is an analytic finitary hypergraph on a Polish spaceY which does not have Borel σ-bounded fractional chromatic number. In thePE-extension of the Solovay model, the chromatic number of G is uncountable.

Proof. In view of Fact 11.1.22 it is enough to consider the case of a large mea-sured skew product G. Find a sequence 〈an, Hn, tn : n ∈ ω〉 which generates Gas in Definition 11.1.21. Thus, Y =

∏n an and G =

∏nHn, tn.

Now, consider the control poset Q: it conditions are finite functions a : X →2<ω such that for some n ∈ ω, rng(a) ⊂ 2n, and for points x0, x1 ∈ dom(a),x0 E x1 iff a(x0) = a(x1). The ordering is defined by b ≤ a if dom(a) ⊂ dom(b)and for all x ∈ dom(a), a(x) ⊂ b(x). The following is the key observation.

Claim 11.6.2. Q is a very Suslin c.c.c. poset. In addition, it is Suslin measured.

Proof. It is clear that Q is very Suslin. To check the very Suslin property of Q,suppose that a ⊂ Q is a countable set. It is not difficult to see that if there isa condition r ∈ Q incompatible with all elements of a, then there is one whosesupport is a subset of

⋃q∈a supp(q). The search for an incompatible condition

among the countably many conditions of this type is a Borel procedure.To verify the Suslin measured property of Q, consider the function m : Q→

(0, 1] defined by m(q) = 2−n2−k where n ∈ ω is such that rng(q) ⊂ 2n, andk ∈ ω is such that dom(q) has nonempty intersection with exactly k many E-classes. For each positive rational number ε > 0 let Aε = {q ∈ Q : m(q) > ε}.Since the sets Aε ⊂ Q are all analytic and Q =

⋃εAε, it is enough to show,

given ε > 0 and a finite sequence 〈qi : i ∈ j〉 of elements of the set Aε, there isa set b ⊂ j of size greater than εj such that the conditions pi for i ∈ b have acommon lower bound.

To this end, let a be the set of all E-classes with nonempty intersectionwith

⋃i dom(qi). Let µ be the usual probability measure on 2ω and let λ be

the product measure on Y = (2ω)a. Define the set B ⊂ j × Y as the set ofall pairs 〈i, y〉 such that ∀x ∈ dom(qi) qi(x) ⊂ y([x]E). By the definition ofthe function m, the vertical sections Bi have λ-mass > ε each. By the Fubinitheorem applied to the counting measure on j and λ, there must be a pointy ∈ Y such that the horizontal section By has size greater than εj. It is easyto check that the set {qi : i ∈ By} has a common lower bound in Q.

Note that a generic filter G ⊂ Q induces a function defined by f(x) =⋃{p(x) : p ∈ G} whenever x ∈ X is a ground model point. It is clear that

dom(f) consists of the ground model elements of X, for all such points x0, x1 ∈X x0 E x1 iff f(x0) = f(x1) holds, and moreover the range of f consist ofelements of 2ω which are not in the ground model.

Now, suppose that p ∈ P is a condition. Consider the poset R which is thefinite support iteration of the posets Q of length ω1. Let fα : X → 2ω be theR-name for the partial function defined by the generic filter on the α-th iterand.Let Mα be the R-name for the model obtained from the first α many iterands.Consider the R-name σ for a function from E-classes to 2ω such that if c is an

11.6. COLLAPSES 289

E-class in dom(p) then σ(c) = p(c), and if c is an E-class not in σ and α ∈ ω1 isthe first ordinal such that c is represented in Mα, then f(c) = y for the uniquey ∈ 2ω such that for all points x ∈ c∩Vα, fα(x) = y. It is not difficult to see thatR forces σ to be an injection of the set of E-classes to 2ω. By Theorem 6.6.3,σ represents a balanced condition in the collapse poset P , and it is not hard tosee that this balanced condition is below p.

Finally, we are ready for the main body of the proof. Let κ be an inaccessiblecardinal, let W be a Solovay model derived from κ, and work in W . Supposethat p ∈ PF is a condition and τ is a P -name such that p τ : Y → ω is afunction. We must find a condition p ≤ p, a number n ∈ ω, and a hyperedgee ∈ G such that for all y ∈ e, p τ(y) = m. The condition p ∈ PE and thename τ are definable from parameters in the ground model and a parameterz ∈ 2ω. Let V [K] be an intermediate extension obtained by a poset of size < κwhich contains the parameter z and work in the model V [K].

Consider the iteration S ∗ R where S is the Cohen poset on the space Y andR is the finite support iteration of the control poset of length ω1. Let ygen be anS-name for the generic point in the space Y , and in the S-extension let σ be theR-name for the balanced condition in PE below p as isolated in the preliminarypart of this proof. Note that the name σ does not depend on ygen per se, but

only on the model generated by ygen . By a standard balance argument, S ∗ R Coll(ω,< κ) ∃m ∈ ω σ ygen ∈ Bm. Thus, let 〈s, r〉 ∈ S ∗ R be a condition

and m ∈ ω be a number such that 〈s, r〉 Coll(ω,< κ) σ ygen ∈ Bm. Now,strengthening s if necessary, we may assume that

• there is a specific countable ordinal α ∈ ω1 such that s r is a conditionin the iteration Rα up to α;

• since the poset Rα is very Suslin and Suslin measured by Theorem 11.3.15,we may find analytic sets An ⊂ Rα and positive rationals εn > 0 for n ∈ ωwitnessing the Suslin measured property of Rα. We may assume that thereis a specific n such that s r ∈ An.

• there is a number k ∈ ω such that 1/k < εn and s = tk.

Now, let y ∈ Y be a point generic over V [K] such that s ⊂ y and work inV [K][y]. For each i ∈ ak let yi ∈ Y be the point obtained from y by rewriting thek-th entry of y with i. In the model V [K][y], for each i ∈ ak let ri = r/yi ∈ Rα.Since ri ∈ An holds for all i ∈ ak, there is a set b ⊂ ak of size at least |ak|/k+ 2such that the conditions {ri : i ∈ b} have nonzero common lower bound. By theinitial assumption on the hypergraph Hk, the set b is not an Hk-anticlique; letc ⊂ b be an Hk-hyperedge. Thus, we have {yi : i ∈ c} ∈ G. Let H ⊂ R be afilter generic over V [K][y] containing all the conditions ri for i ∈ c and p = σ/Hits associated balanced condition. By the forcing theorem, in the model Wp ∀i ∈ c τ(yi) = m. This completes the proof.

Among the many preservation consequences of the theorem, we state the moststriking one.


Corollary 11.6.3. Let P be the collapse poset of E0 to 2ω.

1. In the P -extension of the Solovay model, the chromatic number of thediagonal Hamming graph is uncountable;

2. it is consistent relative to an inaccessible cardinal that ZF+DC holds and|E0| ≤ |2ω| and yet the chromatic number of the diagonal Hamming graphis uncountable; in particular, there is no E0-transversal.

Note that the collapse does change some chromatic numbers to their lowestpossible value; for example, it forces the chromatic number of H2 to be equalto two, by Proposition 11.1.6. The following example shows a somewhat moredramatic case of this behavior, as the hypergraph in question does not haveBorel σ-bounded chromatic number. The example also shows that the use ofthe fractional chromatic number in place of the usual chromatic number is tosome extent necessary in the assumption of the theorem.

Example 11.6.4. Let E be a Borel equivalence relation on a Polish space X.Let Q be the poset of all pairs q = 〈aq, bq〉 of finite subsets of X such that(aq × bq) ∩ E = 0; the ordering is that of coordinatewise reverse inclusion. LetG be the graph connecting two conditions in Q if they are incompatible. Thenin ZF, |E| ≤ |2ω| if and only if the chromatic number of G is countable.

Note that the graph G does not have Borel σ-bounded chromatic number assoon as the equivalence relation E is non-smooth as per Example 11.1.19. Inparticular, there is a Borel graph which does not have Borel σ-bounded chro-matic number, yet the collapse of |E0| to |2ω| forces countable chromatic numberto it.

Proof. For the left-to-right direction, let h : X → 2ω be a reduction of E to theidentity on 2ω. For every n ∈ ω and every set c ⊂ 2n let Bc = {q ∈ Q : ∀x ∈aq h(x) � n ∈ c ∧ ∀x ∈ bq h(x) � n /∈ c}. Since the function h is E-invariant,the sets Bc ⊂ Q are G-anticliques. Now, if q ∈ Q is a condition, there is n ∈ ωsuch that for all x0 ∈ aq and x1 ∈ bq, h(x0) � n 6= h(x1) � n. Thus, writingc = {h(x) � n : x ∈ aq}, we conclude that q ∈ Bc. It follows that Q =

⋃cBc is

a cover of Q by countably many G-anticliques.For the right-to-left direction, letQ =

⋃nBn be a cover ofQ byG-anticliques.

For each n ∈ ω, let An = [⋃q∈Bn aq]E and let h : X → 2ω be the function de-

fined by h(x)(n) = 0 if x ∈ An. We claim that this is a reduction of E tothe identity on 2ω, thus inducing the desired inequality |E| < |2ω|. To seethis, let x0, x1 ∈ X be arbitrary points. If x0 E x1 holds then h(x0) = h(x1)since the sets An are E-invariant. On the other hand, if x0 E x1 fails, thenq = 〈{x0}, {x1} is a condition in Q and there must be n such that q ∈ Bn. It isnot difficult to see that then h(x0)(n) = 0 6= h(x1)(n) and so h(x0) 6= h(x1).

Now we move to the case of a collapse of |E| to |E0| for a Borel pinned equivalencerelation F . Again, let PE be the collapse as defined in Definition 6.6.2. Recallthat its balanced conditions are classified by injections from the E-quotientspace to the E0-quotient space by Theorem 6.6.3.

11.6. COLLAPSES 291

Theorem 11.6.5. Let E be a pinned Borel equivalence relation on a Polishspace X. Let G be a finitary analytic hypergraph on a Polish space Y of un-countable Borel chromatic number. In the PE-extension of the Solovay model,G has uncountable chromatic number.

Proof. Consider the control poset Q: it consists of conditions q such that forsome number nq ∈ ω, q is a finite partial function from X to 2nq . The orderingis defined by r ≤ q if nq ≤ nr, dom(q) ⊆ dom(r), for every x ∈ dom(q),q(x) ⊆ r(x), and for every pair x0, x1 ∈ dom(q) of E-related points and forevery m ∈ nr \ nq, r(x0)(m) = r(x1)(m).

Claim 11.6.6. Q is a very Suslin, Suslin σ-centered poset.

Proof. It is immediate that Q is a Suslin poset. For the very Suslin part,suppose that a ⊂ Q is a countable set. It is not difficult to check that if there isa condition q ∈ Q which is incompatible with all conditions in a, then there mustbe such a condition q with dom(q) ⊂

⋃r∈a dom(r). There are only countably

many conditions satisfying the latter formula, and the search in this countableset for one which is incompatible with all elements of a is a Borel procedure.

For the Suslin σ-centered property of the poset Q, a descriptor of a conditionq ∈ Q is a function h whose domain is a finite collection of pairwise disjoint basicopen subsets of X such that for each O ∈ dom(h) there is exactly one pointxO of dom(q) in O, and for each point of dom(q) belongs to one set in dom(h);moreover, h(O) = q(xO). It is immediate that the set of all conditions with agiven descriptor is Borel and σ-centered, every condition has many descriptors,and there are only countably many possible descriptors in all. Therefore, Q isSuslin σ-centered as required.

Note that the poset Q adds an injection from the ground model E-classesto the collection of E0-classes which are not represented in the ground model.We denote the name for the injection by η; thus η([x]E) is forced to be theE0-class of the point y =

⋃{r(x) : r is in the generic filter}. Now, let R be the

finite support iteration of the poset Q of length ω1, adding functions ηα at eachstage of the iteration; we denote by Vα the generic extension obtained after theα-th stage of the iteration. Let p ∈ PE be any condition. Let σ be the R-namefor the injection from the E-quotient space to the E0-quotient space defined byσ(c) = ap(c) whenever c is an E-class in the domain of ap, and σ(c) = ηα(c)whenever c is an E-class which is not in the domain of ap and α ∈ ω1 is thefirst ordinal such that c has a representative in the model Vα. It is clear that σis a name for an injection from the E-quotient space to the E0-quotient spacecompatible with the condition p.

The rest of the proof follows closely the argument for Theorem 11.4.5 andwe omit it.

Corollary 11.6.7. Let E be a pinned Borel equivalence relation. Let G be ananalytic finitary hypergraph on a Polish space, with uncountable Borel chromaticnumber. It is consistent relative to an inaccessible cardinal that ZF+DC holds,|E| ≤ |E0| and the chromatic number of G is uncountable.


11.7 Compactly balanced posets

In this section we prove an additional preservation property of compactly bal-anced posets of Definition 9.2.1 which concerns the locally countable structures.

Definition 11.7.1. Let G be an analytic finitary hypergraph on a Polish spaceX. The hypergraph is actionable if there is a countable group Γ acting on aPolish space X such that all hyperedges of G consist of pairwise orbit-relatedelements and for every γ ∈ Γ, γ ·G = G.

The following theorem is stated using Convention 1.7.16.

Theorem 11.7.2. Let G be an analytic, finitary, actionable hypergraph on aPolish space X which does not have Borel σ-bounded chromatic number. Thenin compactly balanced extensions of the Solovay model, G has uncountable chro-matic number.

The actionable assumption is necessary, see the example below. The maindifference between the general and actionable hypergraphs we exploit is thefollowing routine strengthening of Fact 11.1.13; instead of skew products wecan deal with the usual product.

Fact 11.7.3. Let G be a finitary analytic actionable hypergraph on a Polishspace X. Exactly one of the following occurs:

1. G has Borel σ-bounded chromatic number;

2. there is a large product H on a Polish space Y and a continuous homo-morphism h : Y → X of H to G.

Proof. In view of Fact 11.7.3, it is enough to prove the theorem for large producthypergraphs. Thus, let 〈an, Hn : n ∈ ω〉 be a sequence such that for every n ∈ ω,Hn is a hypergraph on the finite set an, and |an| ≥ 2 and the chromatic numberof Hn is at least n, and assume that G is in fact the product hypergraph

∏nHn

on the space Y =∏n an. With a large product of this form, we associate the

poset R of all functions r with domain ω such that for every n ∈ ω, r(n) ⊂ an isa nonempty set and the chromatic numbers of Hn on r(n) are unbounded as ntends to infinity. The ordering on R is that of coordinatewise inclusion. Clearly,the poset R adds a point ygen ∈ Y defined by ygen(n) is the only element of theset⋂{r(n) : r belongs to the generic filter}. The following claim is key.

Claim 11.7.4. The poset R is proper, bounding, and adds no independent reals.

Proof. The first two assertions of the following claim are standard and provedby the usual fusion arguments. The last assertion is the heart of the presentproof. Suppose that r ∈ R is a condition and τ is an R-name for a subsetof ω; we have to find a condition s ≤ r and an infinite set c ⊂ ω such thats c ⊂ τ or c ∩ τ = 0. Strengthening the condition r if necessary, a standardfusion argument will yield a continuous function f :

∏n r(n)→ P(ω) such that

r τ = f(ygen).

11.7. COMPACTLY BALANCED POSETS 293

Now, let B ⊂∏n r(n) × ω be the Borel set given by 〈y, n〉 ∈ B if n ∈ f(y)

holds. A partition result [84, Theorem 1.4] applied with an infinite subsequenceof the sets r(n) and the chromatic numbers of the hypergraphs Hn as submea-sures on the sets r(n) shows that there is a condition s ≤ r and an infinite setc ⊂ ω such that either

∏n s(n)× c ⊂ B or (

∏n s(n)× c)∩B = 0. In the former

case, s c ⊂ τ ; in the latter case, s c ∩ τ = 0. The claim has just beenproved.

Let κ be an inaccessible cardinal. Let P be a Suslin poset which is compactlybalanced below κ. Let W be the Solovay model derived from κ and work in themodel W . Let p ∈ P be a condition and τ be a P -name such that p τ : X → ωis a function. We will find a number m ∈ ω, finite sets an ⊂ X for n ∈ ω, anda condition p ≤ p such that for all n, an cannot be covered by n-many G-anticliques, and p ∀x ∈

⋃n an τ(x) = m.

The condition p ∈ P as well as the name τ are definable in the model Wfrom parameters in the ground model and a parameter z ∈ 2ω. Let V [K] be anintermediate forcing extension obtained by a poset of size < κ, containing thepoint z. Work in V [K] and consider the poset Q × R where Q is the poset ofinfinite subsets of ω ordered by inclusion. Let U ⊂ Q and y ∈ Y be mutuallygeneric filter objects for the product; so U is in fact a nonprincipal ultrafilter onω in the model V [K][U ]. The poset Q is σ-closed, and therefore R computed inthe model V [K][U ] is the same as R computed in the model V [K] and y is infact R-generic over the model V [K][U ]. By the continuous reading of names forthe poset R, the sets P(ω) ∩ V [K][U ][y] and P(ω) ∩ V [K][y] are the same. Bya mutual genericity argument and the fact that R adds no independent reals,U generates an ultrafilter on ω in the model V [K][U ][y]. Work in the modelV [K][U ].

Claim 11.7.5. There is an R-name σ for a balanced condition in P such thatC ∀y ∈ Y y = ygen up to finitely many entries→ σ/ygen = σ/γ · ygen .

To parse the claim correctly, note that any point of the space Y which is up tofinitely many entries equal to ygen is in fact R-generic again and yields the sameforcing extension. The claim therefore says that the evaluation of the nameσ does not depend on the specific generic point, but only on its modulo finiteequivalence class.

Proof. Choose an arbitrary R-name χ for a balanced condition below the con-dition p. We now use the ultrafilter U and the compact balance of the posetP to integrate the name χ. Choose any point yω ∈ Y . For each n ∈ ω, let ynbe the R-name below C for the element of Y such that yn � n = yω � n andyn � [n, ω) = ygen � [n, ω); note that yn is forced to be a point R-generic overthe model V [K][U ], generating the same model as ygen . Let χn be the name forthe balanced condition χ/yn. Let σ be the name for the U -limit of the sequence〈χn : n ∈ ω〉 in the definable compact Hausdorff topology on the space of allbalance classes for P . It is immediate that the name σ works as desired.


Still working in the model V [K][U ], find a name σ as in the claim. By astandard balance argument, R Coll(ω,< κ) σ decides in P the value ofτ(ygen). Passing to a condition r if necessary, we may assume that there is aspecific number m ∈ ω such that r Coll(ω,< κ) σ P τ(ygen) = m. Now,let y ∈ Y be a point R-generic over the model V [K][U ], meeting the conditionr and let p = σ/y. Find a number n ∈ ω such that the set r(n) contains an Hn-hyperedge b ∈ Hn and for each i ∈ b let yi ∈ Y be the point obtained from y byreplacing the n-th entry of y with i. Thus, the set {yi : i ∈ b} is a G-hyperedge(this is exactly the point which does not work if a skew product variation of Gis considered) and for each i ∈ n we have p = σ/yi by the choice of the name σ.By the forcing theorem, in the model W p P ∀y ∈ an τ(y) = m, completingthe proof.

Corollary 11.7.6. Let G,H be analytic locally countable hypergraphs on respec-tive Polish spaces X,Y such that G has Borel σ-bounded chromatic number andH is actionable and does not have Borel σ-bounded chromatic number. Thenit is consistent relative to an inaccessible cardinal that ZF+DC holds, G hascountable chromatic number and yet H has uncountable chromatic number.

Proof. Let X =⋃nBn be a partition into Borel sets witnessing the Borel σ-

bounded chromatic number of G. Let E be a countable Borel equivalence rela-tion on X such that all hyperedges of G consist of pairwise E-related elements.For each n ∈ ω let Pn be the poset of functions p whose domain is a countablerelatively E-invariant subset of Bn, range is a subset of n, and p is a partial G-coloring. The ordering on Pn is that of reverse inclusion. Let P =

∏n Pn. Then

we see that in Pn, the balanced conditions are classified by total G-colorings ofthe set Bn whose range is a subset of n. Such colorings naturally form a compactsubset of the compact Hausdorff space nBn , therefore the poset Pn is compactlybalanced. As a result, even the full support product P =

∏n Pn is compactly

balanced, and therefore in the P -extension of the symmetric Solovay model, thechromatic number of H is uncountable by Theorem 11.7.2. At the same time,the poset Pn adds a total G-coloring on the set Bn with n many colors andtherefore the product P forces G to have countable chromatic number.

Corollary 11.7.7. Let Z act freely and in a measure preserving Borel way ona Polish probability space 〈X,µ〉. Let G be the hypergraph of arity three on Xcontaining a triple {x0, x1, x2} if there is a number n ∈ ω such that n · x0 = x1

and n · x1 = x2. Then

1. it is consistent relative to an inaccessible cardinal that ZF+DC holds, thechromatic number of G0 is uncountable and the chromatic number of G iscountable;

2. it is also consistent relative to an inaccessible cardinal that ZF+DC holds,the chromatic number of G0 is two and the chromatic number of G isuncountable.


Proof. For the first item, use Corollary 11.4.20. For the second item, use Ex-ample 11.1.20 to show that the hypergraph G does not have Borel σ-boundedchromatic number; it is clearly actionable. Then, use Theorem 11.7.2 to theposet P adding a G0-coloring with two colors by countable approximationswhose domain is E0-invariant. It is not difficult to see and follows from Theo-rem 6.2.2 that the balanced conditions for P are classified by total G0-coloringswith two colors, which naturally form a closed subset of the compact Hausdorffspace 22ω . Thus the poset P is compactly balanced.

Corollary 11.7.8. 1. Let P be the poset of infinite subsets of ω ordered byinclusion. In the P -extension of the Solovay model, the chromatic numberof H<ω is uncountable.

2. It is consistent relative to an inaccessible cardinal that there is a nonprin-cipal ultrafilter on ω and the chromatic number of H<ω is uncountable.

This is immediate from the fact that the poset P is compactly balanced–Example 9.2.4. The corollary can be viewed as a commentary on a result ofRosendal [76]: in ZF+DC, if there is a discontinuous homomorphism betweenPolish groups then the chromatic numbers of the Hamming graphs Hn are fi-nite. Since a nonprincipal ultrafilter on ω yields a discontinuous homomorphismfrom the Cantor group 2ω to 2, this result cannot be extended to the diagonalHamming graph.

The following two corollaries use compactly balanced posets from Exam-ples 9.2.11 and 9.2.13.


1. Let P be the linearization poset for the E-quotient space as in Exam-ple 8.6.5. In the P -extension of the Solovay model, the chromatic numberof H<ω is uncountable.

2. It is consistent relative to an inaccessible cardinal that there is a linearordering on the E-quotient space and the chromatic number of H<ω isuncountable.

Corollary 11.7.10. Let G be a locally finite Borel graph on a Polish space Xsatisfying the Hall condition.

1. Let P be the poset adding a perfect matching for G as in Example 6.2.4.In the P -extension of the Solovay model, the chromatic number of H<ω isuncountable.

2. It is consistent relative to an inaccessible cardinal that G has a perfectmatching and the chromatic number of H<ω is uncountable.

The following example presents a (necessarily) non-invariant hypergraph whichdoes not have Borel σ-bounded chromatic number, yet it has countable chro-matic number in some compactly balanced extension of the Solovay model.


Example 11.7.11. The actionable assumption cannot be removed from theassumptions of Theorem 11.7.2.

Proof. Let Γ be the free group on two generators γ, δ. It acts on the space 2Γ

by shift; that is, (β · x)(α) = x(β−1α) holds for all β, α ∈ Γ and x ∈ 2Γ. LetX ⊂ 2Γ be the dense Gδ set on which the action is free. Let E0 be the Γ-orbitequivalence relation on X. Let E1 be the orbit equivalence relation induced bythe subgroup of Γ generated by δ. It is not difficult to see that E1 ⊂ E0 are Borelequivalence relations. Let G be the hypergraph on X of arity three containingtriples {x0, x1, x2} consisting of pairwise E0-related points and containing twoE1-related and two E1-unrelated points.

To see how a compactly balanced poset can make the chromatic number ofG countable, consider the Cayley graph H on X: it connects points x0, x1 ifx1 = γ · x0 (the γ-edges of H) or x1 = δ · x0 (the δ-edges of H) or vice versa.This is an acyclic 4-regular graph whose connectedness components are exactlythe E0-classes. In ZFC, every acyclic graph without vertices of degree 0 or 1 hasan orientation in which every vertex has out-degree one, constructed componentby component. The following claim shows the impact of such an orientation onthe hypergraph G:

Claim 11.7.12. (ZF) Suppose that the graph H has an orientation in whichevery vertex has out-degree one. Then the chromatic number of G is countable.

Proof. Let ~H be the orientation. For each x ∈ X the color h(x) is the pair

〈b = b(x), k = k(x)〉 such that k is the largest possible number such that ~Hcontains an oriented path from x to δck · x for some c ∈ {−1, 1}. If k = 0 thenb = 0, otherwise b = c. If such k does not exist,then the color h(x) is ∞. Weclaim that h is a G-coloring.

To see this, consider any two E0-related points x0, x1 ∈ X. In the H-pathbetween x0, x1 no vertex gets an out-degree two, and that leaves only threecases: either the whole path is oriented towards x0, or towards x1, or towardssome point in the middle. Now, if h(x0) = h(x1) =∞ then all three cases showthat all the edges on the path must be δ-edges and so h(x0) E1 h(x1). If x0, x1

are E1-related, say x0 = δm · x1 for some m > 0, and h(x0) 6= ∞, then in thefirst two cases k(x0) 6= k(x1) and in the third case b(x0) = 1 6= −1 = b(x1).It follows that no G-hyperedge can be monochromatic: the homogeneous colorcannot be ∞ on the account of the two E1-unrelated points in the hyperedge,and the homogeneous color cannot be different from ∞ on the account of thetwo E1-related points in the hyperedge.

Claim 11.7.13. The hypergraph G does not have Borel σ-bounded fractionalchromatic number.

Proof. For a point x ∈ X and a number m ∈ ω, write axm = {δimγjm · x : i, j ∈m}. Note that the set axm ⊂ X is a subset of a single E0-class and visits n manydistinct E1-classes, each in n many elements. A simple counting argument thenshows that every subset of axm of size m + 1 contains a G-hyperedge, and so


the fractional chromatic number of G on the set axm is not smaller than m aswitnessed by the normalized counting measure on axm.

Now, suppose that X =⋃nBn is a partition into Borel sets. By the Baire

category theorem, there is a number n ∈ ω such that Bn is not meager. Inview of the first paragraph, to prove the claim it will be enough that for all butfinitely many m ∈ ω there is a point x ∈ X such that axm ⊂ Bn. To this end, lett : Γ→ 2 be a finite partial function such that Bn is comeager in [t]. Let m ∈ ωbe larger than the length of any word in t; we will find a point x ∈ X such thataxm ⊂ Bn. Just let s =

⋃{tij : i, j ∈ m} where tij : Γ → 2 is a finite partial

function given by the demand tij(β) = t(δimγjmβ). By the choice of the numberm, s : Γ → 2 is a finite function. The set C =

⋂{γ−jmδ−imBn : i, j ∈ m} is

comeager in [s]; let x ∈ C be an arbitrary point. Reviewing the definitions, itis clear that axm ⊂ B as required.

Now, consider the poset P adding an orientation of H in which every vertexin X gets an out-degree one. A condition p ∈ P is an orientation on countablymany components of H in which every vertex in these components gets an out-degree one. The ordering is that of reverse inclusion. The poset P was analyzedin Example 9.2.15 It is not difficult to prove that the balanced conditions ofP are classified by orientations of H in which every vertex gets an out-degreeone, which naturally form a compact Hausdorff space. Thus, the poset P iscompactly balanced, and it adds an H-coloring with countably many colors byClaim 11.7.12 while the hypergraph H does not have Borel σ-bounded chomaticor even fractional chromatic number by Claim 11.7.12.


Chapter 12

The Silver divide

12.1 Perfectly balanced forcing

The perfect set property in the model L(R)[U ], where U is a Ramsey ultrafilter,was one of the first results in the literature about models of the type studiedin this book [21]. In this section, we provide a general machinery for provingthe perfect set property type of results, with much less effort than the originalargument quoted above. We start with two key definitions.

Definition 12.1.1. Let P be a Suslin poset. A virtual condition p is perfectlybalanced if in every generic extension V [G], whenever

1. Q ∈ V is a poset such that P(Q) ∩ V is countable in V [G];

2. σ ∈ V is a Q-name for a condition in P stronger than p;

3. H ⊂ PQ is a perfect set such that every finite set a ⊂ H is a set of filtersover Q mutually generic over V ,

then there is a perfect set C ⊂ H such that the set of conditions {σ/H : H ∈ C}has a lower bound in the separative quotient. A poset is perfectly balanced ifbelow every condition p ∈ P there is a virtual perfectly balanced condition.

This forcing property is often guaranteed by a simple feature which does notspeak about any balance issues at all.

Definition 12.1.2. A Suslin forcing P is perfect if for every Borel functionf : 2ω → P , either there is a finite set a ⊂ 2ω such that the set f ′′a ⊂ P hasno lower bound, or else there is a perfect set C ⊂ 2ω such that f ′′C has a lowerbound in the separative quotient.

In general, the separative quotient of Suslin forcings is a Π12 ordering, making

perfectness a rather complicated projective property of the poset P . In allparticular posets considered in this book, the status of perfectness is absolutethroughout all forcing extensions.

299

300 CHAPTER 12. THE SILVER DIVIDE

Proposition 12.1.3. Every Suslin forcing which is perfect in all forcing exten-sions and balanced is perfectly balanced and every balanced virtual condition isperfectly balanced.

Proof. Let p be a balanced virtual condition. Let Q be a partial order, σ aQ-name for a condition in P stronger than p. Let V [G] be a generic extensionsuch that P(Q)∩ V is countable in V [G]. Let H be a perfect set of filters on Qwhich consists of filters in finite tuples mutually generic over V .

Claim 12.1.4. For every finite set a ⊂ H, the set {σ/H : H ∈ a} ⊂ P has acommon lower bound.

Proof. Let 〈Hj : j ∈ i〉 enumerate the set a without repetitions. By induction onj ∈ i, construct a descending sequence 〈rj : j ∈ i〉 of conditions in P such that foreach j ∈ i rj ∈ V [K][Hk : k ∈ j] and rj+1 ≤ σ/Hj . This is easy to do using thebalance of the condition p at every stage of the induction, noting that σ/Hj , rjare both strengthenings of p in mutually generic extensions of the model V [K].In the end, the condition ri is a lower bound of the set {σ/H : H ∈ a}.

Now, let h : 2ω → H be a continuous injection and let f : 2ω → P be thecontinuous function defined by f(y) = σ/h(y). Apply the perfectness of theposet P to find a perfect set C such that for each y ∈ C, r ≤ f(y) holds in theseparative quotient. The condition r witnesses the perfect balance of p in thisinstance.

We prove two notable dichotomy type preservation properties of perfectly bal-anced posets.

Definition 12.1.5. The full Silver dichotomy is the following statement: If Eis a Borel equivalence relation on a Polish space X and A ⊂ X is an E-invariantset, then either A contains only countably many E-classes or A contains a perfectset consisting of pairwise E-unrelated elements.

As a consequence, the full Silver dichotomy implies that among all sets whosecardinality is smaller than a Borel quotient space, 2ω is the set with the smallestuncountable cardinality. The terminology refers to the classical Silver dichotomy[48, Theorem 5.7.1], a theorem of ZF+DC which says in particular that thedichotomy holds for analytic sets A ⊂ X. The Solovay model satisfies the fullSilver dichotomy. The following result is stated using Convention 1.7.16.

Theorem 12.1.6. In the cofinally perfectly balanced extensions of the symmet-ric Solovay model, the full Silver dichotomy holds.

Proof. Let κ be an inaccessible cardinal. Let P be a perfect Suslin forcing suchthat Vκ |= P is balanced in every forcing extension. Let W be a symmetricSolovay model derived from κ and work in the model W . Let p ∈ P be acondition and τ be a P -name such that p τ ⊂ X is an E-invariant setcontaining uncountably many E-classes. The condition p as well as the name τhave to be definable from parameters in V as well as some parameter z ∈ 2ω.

12.1. PERFECTLY BALANCED FORCING 301

Use the assumptions to find an intermediate model V [K] obtained as a genericextension of V by a poset of cardinality smaller than κ such that z ∈ V [K] andP is perfectly balanced in V [K].

Work in the model V [K]. Find a perfectly balanced virtual condition p ≤ p.Note that the equivalence relation E is Borel and therefore it has fewer than

iV [K]ω1 many virtual classes. The cardinality iV [K]

ω1 is countable in the model W ,while the set τ is forced to contain uncountably many E-classes. It follows thatin the model V [K] there has to be a poset R of cardinality smaller than κ, anR-name σ for a condition in P stronger than p and an R-name η for an elementof X which is forced not to be a realization of any virtual E-class in the modelV [K], and R Coll(ω,< κ) σ P η ∈ τ .

Move to some generic extension V [K][G] obtained by a poset of cardinalitysmaller than κ such that P(R)∩V [K] is countable in V [K][G]. Work in V [K][G].Use Proposition 1.7.10 to find a perfect set H of filters on R which are in finitetuples mutually generic over V [K]. Note that by the mutual genericity andthe initial choice of the name η, the points η/H for H ∈ H are pairwise E-unrelated. By the perfect balance of the virtual condition p, there is a perfectset C ⊂ H such that the set {σ/H : H ∈ C} has a lower bound, say q ∈ P inthe separative quotient of P . By the forcing theorem applied in every modelV [K][H] for H ∈ C it is the case that in the model W , the condition q forcesthe perfect set {η/H : H ∈ C} consisting of pairwise E-unrelated elements tobe a subset of τ . The proof is complete.

The second preservation theorem of this section deals with a strong form of thewell-known Open Coloring Axiom, OCA [91].

Definition 12.1.7. OCA+ is the following statement. Whenever X is a Polishspace, A ⊂ X is a set, and Γ is a graph on A which is open in the topologyon A × A inherited from X × X, then either A is a union of countably manyΓ-anticliques, or A contains a perfect Γ-clique.

The following result is stated using Convention 1.7.16.

Theorem 12.1.8. In every cofinally perfectly balanced extension of the sym-metric Solovay model, OCA+ holds.

Proof. Let X be a Polish space and let P be a perfect Suslin forcing. Let κ bean inaccessible cardinal such that in Vκ, in every forcing extension the poset Pis balanced. Let W be a symmetric Solovay model derived from κ. In the modelW , let Γ ⊂ X2 be a symmetric open set, let p ∈ P be a condition, and let τbe a P -name for a subset of X such that p τ cannot be covered by countablymany Γ-anticliques. We must find a perfect set B ⊂ X such that any two pointsof B are Γ-related, and a condition q ≤ p in P which forces B ⊂ τ .

To this end, choose a parameter z ∈ 2ω such that p, τ,Γ are definable fromz. Find an intermediate generic extension V [K] of V by a poset of cardinalitysmaller than κ such that z ∈ V [K] and P is perfectly balanced in V [K]. Workin the model V [K]. Let p ≤ p be a perfectly balanced virtual condition in P .


Since Coll(ω,< κ) p P τ is not covered by countably many Γ-anticliques,and a closure of a Γ-anticlique is still a Γ-anticlique, there must be a poset Rof cardinality smaller than κ, an R-name η for an element of X which belongsto no closed Γ-anticlique coded in V [K], and an R-name σ for a condition in Pstronger than p such that R Coll(ω,< κ) σ P η ∈ τ .

Let V [K][G] be a generic extension obtained by a poset of cardinality smallerthan κ such that P(R) ∩ V [K] is countable in V [K][G]. Work in V [K][G]. Let{Dn : n ∈ ω} enumerate all open dense subsets of finite powers of the poset R inV [K], with infinite repetitions. As in the proof of Theorem 12.2.5, by inductionon |t| build conditions rt ∈ R for t ∈ 2<ω so that

• t ⊂ s implies rs ≤ rt;

• whenever Dn ⊂ Rm is an open dense set for some m < 2n, then everym-tuple of distinct elements from the set {rt : t ∈ 2n} belongs to Dn;

• for all t ∈ 2<ω there are open sets Ot0, Ot1 ⊂ X such that Ot0 ×Ot1 ⊂ Γand rta0 η ∈ Ot0 and rta1 η ∈ Ot1.

In the end, for every binary sequence y ∈ 2ω let Hy ⊂ R be the filter generatedby the conditions {ry�n : n ∈ ω}. Note that any finite tuple of distinct filtersHy for y ∈ 2ω is mutually generic over the model V [K] by the second item inthe inductive construction above. The mutual genericity also shows that thefunction y 7→ η/Hy is a continuous injection from 2ω to X, and its range is aΓ-clique by the third item above. By the perfect balance of the condition p,there is a perfect set C ⊂ 2ω such that the conditions {σ/Hy : y ∈ C} have alower bound q in the separative quotient of the poset P . By the forcing theoremapplied in every model V [K][H] for H ∈ C it is the case that in the model W ,the condition q forces the perfect Γ-clique {η/H : H ∈ C} to be a subset of τ .The proof is complete.

Now it is time for a list of perfect and perfectly balanced forcings.

Example 12.1.9. Let I be an Fσ-ideal on ω. The poset P of all I-positivesubsets of ω ordered by inclusion is perfect.

As a special case, this includes the poset of infinite subsets of ω ordered byinclusion of Section 7.1 and the posets of Section 7.3, both adding ultrafilterson ω with various Ramsey properties.

Proof. Recall that P (I) is the poset of all I-positive subsets of ω, ordered byinclusion. Write I =

⋃n In as a countable union of closed sets, each of which is

closed under taking subset. Let f : 2ω → P(ω) be a Borel function such that forany finite set a ⊂ 2ω,

⋂f ′′a /∈ I holds; we must find a perfect set B ⊂ 2ω such

that the set f ′′B ⊂ P has a lower bound. Thinning the domain of f if necessarywe may assume that the function f is in fact continuous. By induction on n ∈ ωbuild nodes ut ∈ 2<ω for all t ∈ 2n and finite sets bn ⊂ ω such that

• bn /∈ In;

12.1. PERFECTLY BALANCED FORCING 303

• s ⊂ t implies that us ⊂ ut and s is incompatible with t implies us isincompatible with ut;

• for each t ∈ 2n+1 and every y ∈ [ut] it is the case that bn ⊂ f(y).

Once the induction is performed, let b =⋃n bn, let B ⊂ 2ω be the perfect set

of all points y ∈ 2ω such that ∀n∃t ∈ 2n ut ⊂ y and use the continuity of thefunction f to prove that b is the lower bound of the set f ′′B.

To start the induction, let u0 = 0. Now suppose that the nodes ut ∈ 2<ω fort ∈ 2n as well as sets bm for m ∈ n have been constructed. For each t ∈ 2n choosedistinct points yt0, yt1 ∈ [ut] and use the initial assumption on the function fto observe that c =

⋂t∈2n f(yt0) ∩

⋂t∈2n f(yt1) is an I-positive set. Since the

set In ⊂ P(ω) is closed, there must be a finite subset bn ⊂ c which is not inIn. Use the continuity of the function f to find initial segments uta0 ⊂ yt0and uta1 ⊂ yt1 satisfying the second item of the induction hypothesis. Thisconcludes the inductive step.

Corollary 12.1.10. [21] Let P be the poset of infinite subsets of ω ordered byinclusion.

1. In the P -extension of the symmetric Solovay model, the full Silver di-chotomy holds, and OCA+ holds.

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds, thereis a Ramsey ultrafilter on ω, the full Silver dichotomy holds, and OCA+holds.

Example 12.1.11. Let 〈Γ, ·〉 be a countable semigroup. The poset P = P (Γ)of Subsection 7.4 is perfect.

Proof. Recall that elements of P are just sequences in Γω and the ordering isdefined by q ≤ p if there are pairwise disjoint nonempty finite sets an ⊂ ω suchthat q(n) =

∏m∈an p(m). The proof of the perfect property of the poset P is

another fusion argument. Let f : 2ω → P be a Borel function such that for everyfinite set b ⊂ 2ω, the conditions in the set f ′′b have a common lower bound inP ; we must find a perfect set B ⊂ 2ω such that the set f ′′B ⊂ P has a lowerbound.

Thinning the domain of f if necessary we may assume that the function fis continuous. By induction on n ∈ ω build nodes ut ∈ 2<ω for all t ∈ 2n, finitesets at ⊂ ω and elements γn so that

• min(an+1) > max(an);

• s ⊂ t implies that us ⊂ ut and s is incompatible with t implies us isincompatible with ut;

• a0 = 0 and if s ⊂ t then as ∩ at = 0;

• for each t ∈ 2n+1 and each y ∈ [ut], γn =∏m∈at f(y)(m).


Once the induction is performed, let p = 〈γn : n ∈ ω〉, let B ⊂ 2ω be the perfectset of all points y ∈ 2ω such that ∀n ∃t ∈ 2n ut ⊂ y and observe that p is thelower bound of the set f ′′B.

To start the induction, let u0 = 0 and a0 = 0. Now suppose that the nodesut ∈ 2<ω for t ∈ 2n as well as sets as for s ∈ 2≤n and elements γm for m ∈ nhave been constructed. For each t ∈ 2n choose distinct points yt0, yt1 ∈ [ut]and use the initial assumption on the function f to observe that the set c ={f(yt0), f(yt1) : t ∈ 2n} has a lower bound in the poset P . This means that thereare nonempty finite sets ata0, ata1 ⊂ ω disjoint from

⋃s∈2≤n as and a semigroup

element γn ∈ Γ such that∏m∈a

ta0f(yt0)(m) = γn and

∏m∈a

ta1f(yt1)(m) =

γn holds for all t ∈ 2n. Use the continuity of the function f to find initialsegments uta0 ⊂ yt0 and uta1 ⊂ yt1 such that f(yt0)(m) = f(y)(m) holds forall m ∈ at0 and all y ∈ [uta0] and f(yt1)(m) = f(y)(m) holds for all m ∈ at1and all y ∈ [uta1]. This concludes the inductive step.

Corollary 12.1.12. Let P be the poset of Subsection 7.4 designed to add astable ordered union ultrafilter.

1. In the P -extension of the symmetric Solovay model, the full Silver di-chotomy holds, and OCA+ holds.

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds, thereis a stable ordered union ultrafilter on ω, the full Silver dichotomy holds,and OCA+ holds.

Example 12.1.13. The Fin×Fin poset P of subsets of ω × ω with infinitelymany infinite vertical sections, ordered by inclusion, is perfectly balanced butnot perfect.

Proof. We first prove the failure of perfectness. For each n ∈ ω and each binarystring t ∈ 2n, it is easy to find a perfect set At ⊂ P(ω) such that

• At consists of pairwise almost disjoint infinite sets;

• whenever {xt : t ∈ 2n} is a collection such that xt ∈ At, the set⋂t xt ⊂ ω

is infinite.

Then, let ft : [t] → P(ω) be a continuous injection into At for every t ∈ 2<ω.Now, for every y ∈ 2ω let g(y) ⊂ ω × ω be the set of all pairs 〈n,m〉 such thatm ∈ fy�n(y). It is clear that the set g(y) ⊂ ω×ω has all vertical sections infiniteand so is a condition in P . We will show that any finite subset of g′′2ω has alower bound in P , yet no condition in P is below more than countably manyelements of g′′2ω.

For the first assertion, let a ⊂ 2ω be a finite set. Let m ∈ ω be such thatthe strings y � m for y ∈ a are pairwise distinct. Then, for all n ≥ m thesecond item implies that the set

⋂y∈a fy�n(y) is infinite. In other words, the set⋂

y∈a g(y) has all vertical sections beyond m infinite, and so is a lower bound forg′′a. For the second assertion, suppose that p ∈ P is a condition, and assume

12.2. BERNSTEIN BALANCED FORCING 305

towards a contradiction that the set {y ∈ 2ω : p ≤ g(y)} is uncountable. By acounting argument, there must be a number n ∈ ω such that the set {y ∈ 2ω : pnis infinite and modulo finite a subset of g(y)n} is uncountable. Find two distinctelements y0, y1 of the latter set such that y0 � n = y1 � n; denote the commonvalue by t. Then, the intersection ft(y0) ∩ ft(y1) should be finite by the choiceof the function ft, and at the same time it should modulo finite contain theinfinite set pn. This is a contradiction.

Now it is time to show that P is perfectly balanced. Let p ∈ P be a condition.Let a ⊂ ω be an infinite set such that for each n ∈ a, the vertical section anis infinite. Let U be a nonprincipal ultrafilter on ω containing a, and for eachn ∈ ω let Un be a nonprincipal ultrafilter on ω such that if n ∈ a then an ∈ Un.Let p be the virtual condition on a collapse poset, standing for the analytic setA of all sets b ⊂ a such that the set c = {n ∈ ω : bn is infinite} diagonalizes U ,and for each n ∈ c, the set bn diagonalizes Un. The condition p ≤ p is balanced;we will show that it is perfectly balanced.

Let Q be a poset and let σ be a Q-name for a condition in the set A. LetV [G] be a generic extension such that P(Q) ∩ V is countable in V [G] and H isa perfect set of filters on Q in finite tuples mutually generic over V . We mustfind a condition r ∈ P in the model V [G] such that the set {H ∈ H : r ≤ σ/H}is uncountable. ???

12.2 Bernstein balanced forcing

There is a class of posets which is in a precise sense dual to the class of perfectlybalanced forcings.

Definition 12.2.1. Let P be a Suslin poset. A virtual condition p is Bernsteinbalanced if in every generic extension V [G], for every condition p ≤ p, everyinfinite poset Q ∈ V such that P(Q)∩V is countable in V [G], and every perfectfamily H ⊂ P(Q) such that each finite set a ⊂ H is a collection of filtersmutually generic over V , there is a filter H ∈ H such that every condition inP ∩ V [H] below p is compatible with p. The poset P is Bernstein balanced ifthere is a Bernstein balanced virtual condition below every condition in P .

As the simplest initial example, consider the poset of countable partial functionsfrom 2ω to 2 ordered by reverse inclusion. The balanced conditions are classifiedby total functions from 2ω to 2. Each such virtual condition p is in fact Bernsteinbalanced. To see this, note that any condition p ≤ p in any extension V [G] hascountable domain. On the other hand, if H ∈ V [G] is a perfect set of filtersmutually generic over V , then by the product forcing theorem, for each pointx ∈ dom(p) \ V there is at most one filter H ∈ H such that x ∈ V [H]. By acounting argument then, there is a filter H ∈ H such that dom(p)∩V [H]\V = 0and then p is compatible with every condition in V [H] which is stronger thanp.

Bernstein balanced extensions of the Solovay model share a number of regu-larity properties. The main technical tool used in all the theorems below is the


following.

Proposition 12.2.2. Let P be a Suslin poset and let p be a Bernstein balancedcondition in P . Let Q be a partial order and σ a Q-name for a condition in Pstronger than p. Let W be a Solovay model derived from an inaccessible cardinalgreater than κ. In the model W , if H ⊂ P(Q) is a perfect family consisting offilters mutually generic over V , then p forces in P that the set {H ∈ H : σ/Hbelongs to the P -generic filter} is uncountable.

Proof. Work in the model W . Let p ∈ P be a condition stronger than p, anda ⊂ H be a countable set. We must find a filter H ∈ H \ a such that σ/H iscompatible with p. Let V [K] be an intermediate extension obtained by a posetof cardinality smaller than κ containing a code for H, and enumeration of theset a, and the condition Q. Work in the model V [K]. The set H\a is Borel anduncountable, and therefore contains a perfect subset. By the Bernstein balanceof the virtual condition p, there is a filter H ∈ H \ a such that p is compatiblewith all conditions in P ∩V [H] stronger than p, in particular with the conditionσ/H. This concludes the proof.

With the proposition in hand, we begin a line of preservation results. They areall stated using Convention 1.7.16.

Theorem 12.2.3. In cofinally Bernstein balanced extensions of the symmetricSolovay model, there is no finitely additive diffuse probability measure on ω.

Proof. Let κ be an inaccessible cardinal. Let P be a Suslin poset which isBernstein balanced cofinally below κ. Let W be the symmetric Solovay modelderived from κ and work in the model W . Suppose towards a contradictionthat p ∈ P is a condition, τ is a P -name for a map from P(ω) to [0, 1], and pforces τ to be a finitely additive diffuse probability measure. The condition pand the name τ are definable from ground model parameters and an additionalparameter z ∈ 2ω. Find an intermediate model V [K] obtained using a poset ofcardinality smaller than κ such that z ∈ V [K] and P is Bernstein balanced inV [K]. Work in the model V [K].

Let p ≤ p be a Bernstein balanced virtual condition. Let Q be the posetof finite binary strings ordered by reverse end-extensions, with its name x forthe set of those n ∈ ω for which there is a condition q ∈ Q in the genericfilter such that q(n) = 1. There must be a poset R of cardinality smallerthan κ, a condition 〈q, r〉 ∈ P × R, and a Q × R-name σ for a condition in Pstronger than p such that either 〈q, r〉 Coll(ω,< κ) σ P τ(x) ≥ 1/2 or〈q, r〉 Coll(ω,< κ) σ P τ(x) ≤ 1/2. For definiteness, assume that theformer option prevails; with the latter option, replace x with the name for itscomplement and proceed in the same way.

Move to an extension V [K][G] obtained with some poset of cardinalitysmaller than κ such that P(Q × R) ∩ V [K] is countable in V [K][G], and workin the model V [K][G]. Let 〈Om : m ∈ ω〉 be an enumeration of all open densesubsets of all finite powers of Q× R that appear in the model V [K], with infi-nite repetitions. By recursion on m ∈ ω build numbers nm ∈ ω and conditions


〈qit, rit〉 ≤ 〈q, r〉 for i ∈ 3 and t ∈ 2m so that dom(qit) = nm for all i ∈ 3 andt ∈ 2m, and:

• if i ∈ 3 and t ⊂ s are binary strings, then 〈qis, ris〉 ≤ 〈qit, rit〉;

• if Om is an open dense set in the k-fold product of Q×R and k is smallerthan m, then for every i ∈ 3 and every k-tuple 〈tl : l ∈ k of distinctelements of 2m+1, the tuple 〈〈qitl , ritl〉 : l ∈ k〉 belongs to Om;

• whenever i, j are distinct numbers in 3, s, t ∈ 2m and k ∈ nm \ dom(q),then qis(k) and qjt(k) are not both simultaneously equal to 1.

To perform the recursion, start with 〈qi0, ri0〉 = 〈q, r〉 for all i ∈ 3. Now, supposethat nm ∈ ω and conditions 〈qit, rit〉 ≤ 〈q, r〉 for i ∈ 3 and t ∈ 2m have beenfound. First, work on subscript 0: for all t ∈ 2m+1 find conditions 〈q0

0t, r0t〉 ∈Q × R so that the first two items are satisfied and let n0m = maxt dom(q0

0t).Then, work on subscript 1: for all t ∈ 2m+1 find conditions 〈q0

1t, r1t〉 ∈ Q × Rso that the first two items are satisfied, for each t and each j ∈ [nm, n0m)q01t(j) = 0 holds, and let n1m = maxt dom(q0

1t). Finally, work on subscript2: for all t ∈ 2m+1 find conditions 〈q0

2t, r2t〉 ∈ Q × R so that the first twoitems are satisfied, for each t and each j ∈ [nm, n1m) q0

2t(j) = 0 holds, and letn2m = maxt dom(q0

2t). To conclude the work, extend the binary strings q00t, q

01t,

and q02t with zeroes only so that the resulting binary strings q0t, q1t, q2t have the

same domain nm+1.In the end, for each i ∈ 3 and each y ∈ 2ω let Hiy ⊂ Q × R be the filter

generated by the conditions 〈qiy�m, riy�m〉 for m ∈ ω. The first and second itemsabove show that the family Hi = {Hiy : y ∈ 2ω} consists of filters on Q× R infinite tuples mutually generic over the model V [K]. The third item above showsthat for distinct i, j ∈ 3 and points y0, y1 ∈ 2ω, the intersection x/Hiy0 ∩ x/Hjy1

is finite.Now, a twice repeated use of the Bernstein balance of the virtual condition

p in V [K] with the families H0,H1, H2 yields points y0, y1, y2 ∈ 2ω such thatthe conditions p0 = σ/H0y0 , p1 = σ/H1y1 , and p2 = σ/H2y2 have a commonlower bound. Write x0 = x/H0y0 , x1 = x/H1y1 , and x2 = H2y2 . By theprevious paragraph, the sets x0, x1, x2 ⊂ ω are pairwise almost disjoint. At thesame time, by the forcing theorem in V [K], the common lower bound of theconditions p0, p1, p2 forces the numbers τ(x0), τ(x1), and τ(x2) to be at least1/2 each. This contradicts the assumption that τ was forced to be a diffusefinitely additive measure.

The next preservation theorem deals with the well-known Open Coloring Axiom,OCA [91].

Definition 12.2.4. OCA is the following statement. Whenever X is a Polishspace, A ⊂ X is a set, and Γ is a graph on A which is open in the topologyon A × A inherited from X × X, then either A is a union of countably manyΓ-anticliques, or A contains an uncountable Γ-clique.


Theorem 12.2.5. In cofinally Bernstein balanced extensions of the symmetricSolovay model, OCA holds.

Proof. Let κ be an inaccessible cardinal. Let P be a Suslin forcing such that Pis Bernstein balanced cofinally below κ. Let W be a symmetric Solovay modelderived from κ. In the model W , let X be a Polish space and let Γ ⊂ X2 bea symmetric open set, let p ∈ P be a condition, and let τ be a P -name for asubset of X such that p τ cannot be covered by countably many Γ-anticliques.We will find a perfect set C ⊂ X such that any two points of C are Γ-related,and a condition q ≤ p in P which forces C ∩ τ is uncountable.

To this end, choose a parameter z ∈ 2ω such that p, τ,Γ are definable fromz. Find an intermediate generic extension V [K] of V by a poset of size < κsuch that z ∈ V [K] and P is Bernstein balanced in V [K]. Work in the modelV [K]. Let p ≤ p be a Bernstein balanced virtual condition. Since Coll(ω,<κ) p P τ is not covered by countably many Γ-anticliques, and a closureof a Γ-anticlique is still a Γ-anticlique, there must be a poset R of size < κ,and R-name η for an element of X which belongs to no closed Γ-anticliquecoded in V [K], and an R-name σ for a condition in P stronger than p such thatR Coll(ω,< κ) σ P η ∈ τ .

Move into the model W . Let {Dn : n ∈ ω} enumerate all open dense subsetsof finite powers of the poset R in V [K], with infinite repetitions. By inductionon |t| build conditions rt ∈ R for t ∈ 2<ω so that

• t ⊂ s implies rs ≤ rt;

• whenever Dn ⊂ Rm is an open dense set for some m < 2n, then everym-tuple of distinct elements from the set {rt : t ∈ 2n} belongs to Dn;

• for all t ∈ 2<ω there are open sets Ot0, Ot1 ⊂ X such that Ot0 ×Ot1 ⊂ Γand rta0 η ∈ Ot0 and rta1 η ∈ Ot1.

The construction is routine except for the last item; we just describe how thelast item is obtained. Suppose that rt has been found. Work in V [K] and letU ⊂ X be the union of all basic open sets O ⊂ X such that rt η /∈ O. Thenrt η ∈ X \ U . Since the set X \ U is closed, by the choice of the nameη it cannot be a Γ-anticlique and therefore there are points x0, x1 ∈ X \ Uwhich are Γ-connected. Since the set Γ ⊂ X2 is open, there are basic open setsOt0, Ot1 ⊂ X such that x0 ∈ Ot0, x1 ∈ Ot1, and Ot0 × Ot1 ⊂ Γ. Neither of thetwo open sets is a subset of U and therefore there must be conditions rta0 andrta1 below rt such that the former forces η ∈ Ot0 and the latter forces η ∈ Ot1as desired.

In the end, for every binary sequence y ∈ 2ω let Hy ⊂ R be the filtergenerated by the conditions {ry�n : n ∈ ω} and let py = σ/Hy. Note that anyfinite tuple of distinct filters Hy for y ∈ 2ω is mutually generic over the modelV [K] by the second item in the inductive construction above. Proposition 12.2.2applied in V [K] shows that in W , p forces the set a = {y ∈ 2ω : py is in thegeneric filter} to be uncountable. The set {η/Hy : y ∈ a} is then forced to bean uncountable Γ-clique.


Theorem 12.2.6. Let E be a Borel equivalence relation on a Polish spaceX. In cofinally Bernstein balanced extensions of the symmetric Solovay model,every subset of X is either covered by countably many E-classes or contains anuncountable subset consisting of pairwise E-unrelated elements.

Proof. Let κ be an inaccessible cardinal. Let P be a Suslin forcing which isBernstein balanced cofinally below κ. Let W be the symmetric Solovay modelderived from κ, let p ∈ P be a condition and let τ be a P -name such thatp τ ⊂ X is a set which is not covered by countably many E-classes. Wewill find a perfect set A ⊂ X consisting of pairwise E-unrelated elements and acondition q ≤ p, q τ ∩ A is uncountable. This will prove the theorem.

The condition p and the name τ are definable from a parameter z ∈ 2ω

and some parameters in the ground model. Let V [K] be some intermediateextension of the ground model by a poset of size < κ containing z and suchthat P is Bernstein balanced in V [K]; work in V [K]. Let p ≤ p be a Bernsteinbalanced condition in the poset P . Since the equivalence relation E is Borel,there are fewer than iω1

many virtual E-classes in V [K]. Since the cardinal iω1

is countable in W , there must be in V [K] a poset R of size < κ and R-namesσ for a condition in P stronger than p and η for an element of X which is nota realization of any virtual E-class in V [K] such that R Coll(ω,< κ) σ Pη ∈ τ .

In the model W , use Proposition 1.7.10 to find a perfect set {Hy : y ∈ 2ω} offilters on R which are mutually generic over the model V [K]. Write py = σ/Hy

and xy = η/Hy. By the assumption on the name η and the mutual genericity,the set A = {η/Hy : y ∈ 2ω} consists of pairwise E-unrelated elements. Propo-sition 12.2.2 applied in V [K] shows that in W the condition p forces that foruncountably many y ∈ 2ω, py belongs to the generic filter. Then p τ ∩ A isuncountable as required.

Theorem 12.2.7. Let I be an analytic P-ideal on ω. In cofinally Bernsteinbalanced extensions of the symmetric Solovay model, if A ⊂ I is an uncountableset, then there is a set b ∈ I such that the set {a ∈ A : a ⊂ b} is uncountable.

Proof. Suppose that κ is an inaccessible cardinal and P is a Suslin poset whichis Bernstein balanced cofinally below κ. Let W be a symmetric Solovay modelderived from κ and in the model W , let p ∈ P be a condition and τ be a P -namesuch that p τ ⊂ I is an uncountable set. We need to produce a conditionp ≤ p and a set b ∈ I such that q {a ∈ τ : a ⊂ b} is uncountable. Thecondition p and the name τ must be definable from parameters in the groundmodel and some real parameter z ∈ 2ω. Let V [K] be an intermediate forcingextension by a poset of size < κ such that z ∈ V [K] and P is Bernstein balancedin V [K], and work in the model V [K].

By the assumptions on the poset P , there must be a Bernstein balancedvirtual condition p ≤ p in the model V [K]. There must be a poset Q of size< κ, a Q-name η for an element of I which is not in V [K], and a Q-name σfor a condition in P stronger than p such that Q Coll(ω,< κ) σ P η ∈ τ .Let V [K][G] be some generic extension by a poset of cardinality less than κ in


which there is a perfect collection H ⊂ P(Q) which consists of filters mutuallygeneric over V [K]. Work in the model V [K][G].

Use the result of Solecki [85] to fix a lower-semicontinuous submeasure µ onω such that I = {a ⊂ ω : limn µ(a \ n) = 0}. Consider the poset R of triples〈s, ε, a〉 where s ⊂ ω is finite, ε > 0 is a real number, and a ∈ I is a set such thatµ(a \ s) < εq. The ordering is defined by 〈s1, ε1, a1〉 ≤ 〈s0, ε0, a0〉 if s0 ⊂ s1,ε1 ≤ ε0, and a0 ⊂ s1 ∪ a0. It is not difficult to see that R is a σ-linked poset,and the union of the first coordinates of conditions in the generic filter is forcedto modulo finite contain every ground model element of I.

Let a ⊂ ω be a set generic over the model V [K][G] for the poset R, and workin the model V [K][G][a]. For every filter H ∈ H ∩ V [K][G], η/H ⊂ a modulofinite holds, and the set H ∩ V [K][G] is uncountable as the poset R is c.c.c.Thus, the set {H ∈ H : η/H ⊂ a modulo finite} is analytic, uncountable, andtherefore contains a nonempty perfect subset H′. Proposition 12.2.2 applied inV [K] shows that in the model W the condition p forces the set A = {H ∈ H′ : σyis in the generic filter} to be uncountable. The set b = {η/H : H ∈ A} is thenforced to be an uncountable subset of τ and each element of it is modulo finiteincluded in the set a.

Now we move to a rich list of examples of Bernstein balanced forcings andrelated corollaries.

Example 12.2.8. Every placid Suslin forcing P is Bernstein balanced and everyplacid virtual condition in P is Bernstein balanced.

Proof. Let p be a virtual placid condition in the poset P . Let V [G] be a genericextension and in V [G], let p ≤ p be a condition in P , let Q ∈ V be an infiniteposet such that P(Q) ∩ V is countable in V [G], and suppose that H ⊂ P(Q) isa perfect family of filters in finite tuples mutually generic over V . We must finda filter H ∈ H such that all conditions in P ∩V [H] below p are compatible withP . By a Mostowski absoluteness argument, it is enough to find such a filter insome further generic extension of V [G]. Let R be any poset adding a new real,and let τ be an R-name for an element of H which is not in V [G].

Claim 12.2.9. R V [τ ] ∩ V [G] = V .

Proof. Suppose towards a contradiction that this is not the case. Then in V [G],there must be a condition r ∈ R, a set a /∈ V of ordinals, and a Q-name η suchthat r a = η/τ . Let K0,K1 ⊂ R be filters mutually generic over V [G], andlet H0 = τ/K0, H1 = τ/K1. Since in the model V [G], the perfect family Hconsisted of filters on Q mutually generic over V , by a Mostowski absolutenessargument this is also true in V [G][K0][K1]; in particular, H0, H1 ⊂ Q are filtersmutually generic over V . By the product forcing theorem applied in V then,V [H0] ∩ V [H1] = V . However, the set a /∈ V belongs to both V [H0] andV [H1] as a = η/H0 = η/H1 by the initial assumptions on the set a. This is acontradiction.


Now, let K ⊂ R be a filter generic over the model V [G] and let H = τ/K; by theclaim, V [G] ∩ V [H] = V . The placidity of the virtual condition p applied withthe models V [G] and V [H] now shows that p is compatible with all conditionsinthe model V [H] below p as desired.

Corollary 12.2.10. Let X be a Polish vector space over a countable field F .

1. Let P be the poset of countable subsets of X which are linearly independentover F , with the reverse inclusion ordering. In the P -extension of thesymmetric Solovay model, there is no nonprincipal ultrafilter on ω andOCA holds.

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds, Xhas a linear basis over F , there is no nonprincipal ultrafilter on ω andOCA holds.

Example 12.2.11. Let Γ be a Borel graph on a Polish space X such thatfor some n ∈ ω, Γ does not contain an injective homomorphic copy of Kn,ω1

.Then the coloring poset PΓ of Definition 8.1.1 is Bernstein balanced and everybalanced virtual condition is Bernstein balanced.

Proof. The balanced virtual conditions are classified by Γ-colorings c : X → ω byTheorem 8.1.2. We must prove that every such a coloring represents a Bernsteinbalanced condition.

To this end, let V [G] be a generic extension and work in V [G]. Towardscontradiction, suppose that there is a condition p ∈ P such that c ⊂ p, an infiniteposet Q ∈ V such that P(Q) ∩ V is countable in V [G], and an uncountablefamily H consisting of filters on Q which are mutually generic over V such thatfor each H ∈ H there is a condition pH ∈ P ∩ V [H] such that c ⊂ pH and pHis incompatible with p. An examination of the incompatibility options and acounting argument reveal that there must be a point x ∈ dom(p) \ V such thatthe family G = {H ∈ H : ∃xH ∈ X ∩ V [Hy] \ V x Γ xH} is uncountable.

Let a0, a1 ⊂ G be disjoint sets of size n. By the initial mutual genericitydemand, the generic extensions M0 and M1 obtained from V by attaching allfilters in the set a0 and a1 respectively are mutually generic extensions of V . Bythe forcing theorem, the point x ∈ X \V belongs to at most one of them; say itdoes not belong to the model M0. For each H ∈ a0, let xH ∈ X ∩V [H]\V be apoint Γ-related to x; by the product forcing theorem, these points are pairwisedistinct. By the initial assumptions on the graph Γ, the set B = {z ∈ X : ∀H ∈a0 z Γ xH} is countable. By the Shoenfield absoluteness between the modelsM0 and V [G], B ⊂ M0 must hold. At the same time, x is an element of Bwhich does not belong to the model M0. A contradiction.

Example 12.2.12. Let K be a Gδ matroid on a Polish space X and P bethe poset of countable K-sets as in Definition 6.5.1. The poset P is Bernsteinbalanced and every balanced virtual condition is Bernstein balanced.


Proof. The balanced virtual conditions are classified by maximal K-sets by The-orem 6.5.2. We must prove that every such a maximal set A ⊂ X represents aBernstein balanced condition.

To this end, let V [G] be a generic extension and work in V [G]. Towardscontradiction, suppose that there is a condition p ∈ P such that A ⊂ p, an infi-nite poset Q ∈ V such that P(Q)∩V is countable in V [G], and an uncountablefamily H consisting of filters on Q which are mutually generic over V such thatfor each H ∈ H there is a condition pH ∈ P ∩ V [H] such that A ⊂ pH and pHis incompatible with p. An examination of the incompatibility options and acounting argument reveal that there must be a finite set b ⊂ p\V such that thefamily G = {H ∈ H : there is a finite set aH ⊂ X such that aH ∪ A is a K-setwhile aH ∪A ∪ b is not} is uncountable.

Let |b| = n. Let d ⊂ G be a set of size greater than n. Let c ⊂ A be a finiteset such that for every filter H ∈ d, b ∪ c ∪ aH /∈ K holds. Note that b ∪ c ∈ Kand (by a repeated application of the balance of the set A)

⋃H∈d aH ∪ c ∈ K.

Let e = b ∪ c ∪⋃H∈d aH and let |e| = m. Since

⋃H∈d aH ∪ c ⊂ e is a set in K

of cardinality m − n, the exchange property of the matroid K guarantees thatevery subset of e maximal with respect to membership in K has cardinality atleast m − n. Let f ⊂ e be a maximal set in K extending the set b ∪ c. Since|f | ≥ m − n, there must be a filter H ∈ d such that aH ⊂ f . This contradictsthe assumption that b ∪ c ∪ aH /∈ K.

Corollary 12.2.13. Let X be a Polish field and F ⊂ X a countable subfield.

1. Let P be the poset of countable subsets of X which are algebraically inde-pendent over F , with the reverse inclusion ordering. In the P -extensionof the symmetric Solovay model, there is no nonprincipal ultrafilter on ωand OCA holds.

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds, Xhas a transcendence basis over F , there is no nonprincipal ultrafilter on ωand OCA holds.

Many quotient simplicial complex forcings are Bernstein balanced.

Example 12.2.14. Let F be a Fraisse class with strong amalgamation. LetE be a Borel equivalence relation on a Polish space X. Then the E,F-Fraisseforcing of Definition 8.6.3 is Berstein balanced and every balanced condition isBernstein balanced.

Proof. The balanced conditions are classified by Theorem 8.6.4 as the F-structureson the virtual E-quotient space. Now, let p be such a balanced virtual condi-tion. Move to some generic extension and in it, suppose that there is a conditionp ∈ P such that p ≤ p, an infinite poset Q ∈ V such that P(Q)∩V is countablein V [G], and an uncountable family H consisting of filters on Q which are mu-tually generic over V . By the mutual genericity, the only E-classes that occurin more than one model V [H] for H ∈ H are the classes which are realizationsof virtual E-classes in V . By a counting argument, there is a filter H ∈ H such


that in the model V [H] there are no E-classes represented in the domain of pother than the classes which are realizations of virtual E-classes in V . Then pis compatible with every condition in V [H] which is stronger than p.


1. Let P be the poset of linear orders on countable subsets of the E-quotientspace, with the reverse inclusion ordering. In the P -extension of the sym-metric Solovay model, there is no nonprincipal ultrafilter on ω and OCAholds.

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds, theE-quotient space is linearly ordered, there is no nonprincipal ultrafilter onω and OCA holds.

There are many posets which are neither Bernstein balanced nor perfectly bal-anced. The examples below are built to violate specific preservation propertiesof Bernstein balanced posets.

Example 12.2.16. Consider the clopen graph Γ on 2ω connecting points x0 6=x1 if the smallest n ∈ ω such that x0(n) 6= x1(n) is even. Let P be the balancedposet of Example 8.8.3. In the P -extension of the Solovay model, there is anuncountable subset of 2ω such that every Γ-clique and every Γ-anticlique in it iscountable. Thus, in the P -extension of the Solovay model, OCA fails. In viewof Theorem 12.2.5, P is not Bernstein balanced.

Example 12.2.17. Let P be the Lusin poset of Definition 8.8.5. In the P -extension of the symmetric Solovay model in which the conclusion of Theo-rem 12.2.6 fails. In particular, the Lusin poset is not Bernstein balanced.

Proof. Let κ be an inaccessible cardinal, let W be a symmetric Solovay modelderived from κ, and move to the model W . Let p = 〈a, b〉 ∈ P be a conditionand let τ be a P -name for an uncountable subset of A. We must find a conditionq ≤ p and two distinct E0-related points x0, x1 ∈ 2ω such that q x0, x1 ∈ τ .To this end, find a parameter z ∈ 2ω such that both p, τ are definable fromsome parameters in the ground model and the parameter z. Let V [K] be anintermediate generic extension containing the parameter z, obtained as a forcingextension of the ground model by a poset of size less than κ. Work in the modelV [K].

Let p ≤ p be a balanced virtual condition in P as in Theorem 8.8.6, soap = a. Since τ is forced to be an uncountable set, there must be a poset R, anR-name σ for a condition in P stronger than p, and an R-name η for a point inaσ \ a such that R Coll(ω,< κ) σ P η ∈ τ . Note that η is forced to be apoint Cohen-generic over the model V [K], and it is also forced that any tuplex ∈ (2ω)n of points in the set aσ \ (a ∪ [η]E0

) is Cohen-generic over the modelV [K][η]. By the general forcing theory, we may present the poset R as a twostep iteration S0 ∗ S1, where S0 is the Cohen poset 2<ω restricted to some finite


binary string s, adding the Cohen-generic point η, and S1 is the S0-name forthe remainder forcing.

Move back to the model W . Choose a filter H00 ⊂ S0 generic over the model

V [K], and write x0 = η/H00 ∈ 2ω. Flip any value of x0 past dom(s) to obtain

another point x1 ∈ 2ω. Then x1 is E0-related to x0 and also Cohen generic overthe model V [K], containing the string s as an initial segment. Write H0

1 ⊂ S0

for the filter generic over V [K] obtained from x1; thus, V [K][H00 ] = V [K][H0

1 ].Find filters H1

0 ⊂ S1/H00 and H1

1 ⊂ S/H01 mutually generic over the model

V [K][H00 ], and write p0 = σ/H0

0 ∗ H10 and p1 = σ/H0

1 ∗ H11 . The conditions

p0, p1 are compatible in the poset P by a mutual genericity argument identicalto Claim ??. Let q ∈ P be any lower bound for the conditions p0, p1. The initialchoices of the names σ and η now imply that q x0, x1 ∈ τ as desired.

Example 12.2.18. Let P be the balanced poset of Example 8.8.4. In the P -extension of the Solovay model there is a dominating subset of ωω such that theset of all functions in in dominated by any fixed y ∈ ωω is countable. Thus, theconclusion of the theorem fails in the P -extension with the P-ideal I on ω × ωconsisting of sets with all vertical sections finite. The poset P is not Bernsteinbalanced.

We conclude this section with an anti-preservation result and an example justi-fying the choice of the Berstein name for the class of posets in question. Recallthat a Bernstein set is a subset of a Polish space such that neither it nor itscomplement contain a perfect set.

Proposition 12.2.19. In nontrivial, cofinally Bernstein balanced extensions ofthe symmetric Solovay model, there is a Bernstein set.

Proof. Suppose that κ is an inaccessible cardinal and P is a Suslin partial orderwhich is Bernstein balanced cofinally below κ. Let W be the symmetric Solovaymodel derived from κ and work in the model W . Let p ∈ P be a condition.Let V [K] be an intermediate extension obtained with a poset of size < κ suchthat p ∈ V [K] and P is Bernstein balanced in V [K]. Working in V [K], find aBernstein balanced condition p ≤ p, and a poset Q of size < κ and a Q-name σfor a condition such that Q σ ≤ p. Since the poset P contains no atoms, itis also possible to find Q-names σ0, σ1 for incompatible elements of P strongerthan σ.

Back in W , let {Hy : y ∈ 2ω} be a perfect collection of filters on Q pairwisegeneric over the model V [K], obtained by an application of Proposition 1.7.10.Let τ0 be the P -name for the set {y ∈ 2ω : σ0/Hy belongs to the generic fil-ter on P} and let τ1 be the P -name for the set {y ∈ 2ω : σ1/Hy belongs tothe generic filter on P}. These are clearly forced to be disjoint subsets of 2ω,and p forces that complement of neither contains a perfect set: any condi-tion below p is incompatible with only countably many conditions in the set{σ0/Hy, σ1/Hy : y ∈ 2ω} by the Bernstein balance of the virtual condition p.Thus, p forces both τ0, τ1 to be Bernstein sets.

12.3. PLACID FORCING 315

Unlike the perfectly balanced extensions, the Bernstein balanced extensions mayexhibit chaotic structure of cardinalities below 2ω. This concern was addressedin [97].

Example 12.2.20. Let P be the poset of all countable functions from 2ω to 2,ordered by reverse extension. Let κ be an inaccessible cardinal, let W be thesymmetric Solovay model derived from κ, and let G ⊂ P be a filter generic overW . In W [G], let a0 = {y ∈ 2ω : ∃p ∈ G p(y) = 0} and a1 = {y ∈ 2ω : ∃p ∈G p(y) = 1}. In W [G], |a0| 6= |a1|.

Thus, 2ω can be decomposed into two uncountable sets of distinct cardinalities,contradicting the full Silver dichotomy. In fact, one can prove all kinds ofpathologies regarding the cardinalities of a0 and a1 respectively; for example|a2

0| 6≤ |a0| holds in W [G].

Proof. Work in W . Suppose towards a contradiction that p ∈ P is a conditionand τ is a P -name such that p τ : a0 → a1 is a bijection. The condition p aswell as the name τ have to be definable from some ground model parametersand some parameter z ∈ 2ω. Find an intermediate model V [K] obtained as ageneric extension of the ground model by a poset of size < κ such that z ∈ V [K].

Work in the model V [K]. Let p be the Coll(ω, c)-name for the set {q ∈P : p ⊂ q ∧ ∀x ∈ V [K] ∩ 2ω \ dom(p) q(x) = 0}. This is a balanced virtualcondition in P for the model V [K]. For every point x ∈ 2ω \dom(p) let yx ∈ 2ω

be a point such that Coll(ω,< κ) p P τ(x) = yx, if it exists. As τ is forcedto be an injection from a0 to a1, for each x there can be at most one yx of thiskind, and the function g : x 7→ yx must be an injection from 2ω \ dom(p) todom(p). Since the former set is uncountable and the latter is countable, theremust be a point x ∈ 2ω \ dom(p) which is not in the domain of g.

Still in the model V [K], the statement that x /∈ dom(g) means that Coll(ω,<κ) forces that either there is a condition below p in P which forces τ(x) out ofV [K], or there are two distinct conditions in P stronger than p which force thevalue τ(x) to be two distinct points in V [K]. Suppose for definiteness that theformer is the case. Then there must be a poset R of size < κ, an R-name σ fora condition in P stronger than p and an R-name η for an element of 2ω \ V [K]such that R Coll(ω,< κ) σ P τ(x) = η.

In the model W , pick filters H0, H1 ⊂ R mutually generic over the modelV [K]. Let p0 = σ/H0 ∈ P and p1 = σ/H1 ∈ P ; by the balance of the conditionp, p0 and p1 are conditions compatible in P , with some lower bound q ∈ P . Lety0 = η/H0 ∈ 2ω and y1 = η/H1 ∈ 2ω; by the product forcing theorem and thechoice of the name η, y0 6= y1 holds. But then, q τ(x) = y0 and τ(x) = y1, animpossibility.

12.3 Placid forcing

Placid forcings have been introduced in Definition 9.3.1; as we have seen inExample 12.2.8, they are Bernstein balanced. In this section, we prove a couple


of preservation theorems for placid forcing which show among other things thatthe matroid posets in Example 12.2.12 and the coloring posets in 12.2.11 arenot placid. Thus, the placid forcings form a fairly small and tightly controlledsubclass of Bernstein balanced forcings.

Theorem 12.3.1. Let X be an uncountable Polish field and F ⊂ X be a count-able subfield. In cofinally placid extensions of the symmetric Solovay model,there is no transcendence basis of X over F .

Proof. Let κ be an inaccessible cardinal. Let P be a Suslin poset which is placidcofinally below κ. Let W be the symmetric Solovay model derived from κ, andwork in W . Suppose towards contradiction that there is a condition p ∈ P anda P -name τ such that p forces τ to be a transcendence basis of X over F . Thename τ and the condition p are definable from ground model parameters andan additional parameter z ∈ 2ω. Let V [K] be an intermediate model obtainedby a forcing of cardinality smaller than κ such that z ∈ V [K] and P is placidin V [K]. Work in V [K].

Let p ≤ p be a placid virtual condition. Consider the closed set Y ={〈x0, x1, x2, x3〉 ∈ X4 : x0x2 + x1x3 = 1} equipped with the topology inher-ited from X4, and consider the associated Cohen poset PY . Theorems 3.2.7and 3.1.4 show that the poset PY adds a quadruple 〈xi : i ∈ 4〉 of points inX such that x0, x1 are mutually Cohen generic elements of X, so are x2, x3,moreover V [K][x0, x1] ∩ V [K][x2, x3] = 0 and x0x2 + x1x3 = 1. For each i ∈ 4,the model V [K][xi] contains a poset Ri of cardinality smaller than κ, an Ri-name σi for a condition in P stronger than p, and a name ηi for a finite setsuch that the entries of the vector xi is forced to be algebraic over ηi andRi Coll(ω,< κ) σi P ηi ⊂ τ . Let Hi ⊂ Ri for i ∈ 4 be filters mutuallygeneric over the model V [K][xi : i ∈ 4]. Write pi = σi/Hi and ai = ηi/Hi.

Claim 12.3.2. The conditions pi for i ∈ 4 have a common lower bound.

Proof. First, p0, p1 have a lower bound p01 ∈ V [K][x0, x1][H0, H1] by the bal-ance of p. Similarly, p2, p3 have a lower bound in V [K][x2, x3][H2, H3] by thebalance of p again. Finally, V [K][x0, x1][H0, H1]∩V [K][x2, x3][H2, H3] = V [K]by the initial choice of the points xi for i ∈ 4 and the mutual genericity of thefilters Hi for i ∈ 4. The placidity of the virtual condition p then shows that p01

and p23 have a common lower bound.

Claim 12.3.3. The set⋃i ai is not algebraically free.

Proof. By the exchange property of the algebraic matroid, there is a point y ∈a3 \ V which is in the algebraic closure of (a3 ∪ {x3}) \ {y}. Since y is not anelement of V , it is also not an element of a0 ∪ a1 ∪ a2. Since x3 is algebraic over{x0, x1, x2} and each of these points are in turn algebraic over a0, a1, and a2, itfollows that y is in the algebraic closure of a0 ∪ a1 ∪ a2 ∪ (a3 \ {y}). This provesthe claim.

Thus, the lower bound of the conditions pi for i ∈ 4 identifies a finite non-freeset⋃i ai ⊂ X forced to be a subset of τ . This is a contradiction.


To state the following theorem in full generality, consider a definition.

Definition 12.3.4. A curve graph is a Borel graph Γ on R2 if there is a Borelset A ⊂ R2 consisting of graphs of some collection of smooth curves in R2

which are closed as subsets of R2 and contain no straight segments, such that Aaccumulates around 0 and Γ = {{y0, y1} ∈ [R2]2 : y0 − y1 ∈ A or y1 − y0 ∈ A}.

One good example of a curve graph is obtained as follows. Let B ⊂ R be a set ofpositive reals accumulating around 0. Let A be the collection of circles aroundthe origin with radii belonging to B. Thus, the resulting curve graph Γ consistsof pairs of points whose Euclidean distance belongs to B. If B is countable,then Γ does not contain an injective homomorphic copy of K2,ω1

and thereforehas countable coloring number by [25]. This is in effect Example 8.1.8.

Another example is obtained by choosing a closed smooth curve C containingno straight segments and passing though the origin such that for some numbern ∈ ω, n many distinct points in the plane belong to at most one translate ofC–perhaps a circle through the origin with n = 3 comes to mind. Let A = C.The resulting curve graph Γ does not contain an injective homomorphic copyof Km,m whenever m ∈ ω is a number large enough so that

(mm

)→(nn

)2

holds;therefore, it has countable coloring number by [25].

Theorem 12.3.5. Let Γ be a curve graph on R2. In cofinally placid extensionsof the Solovay model, the chromatic number of Γ is uncountable.

Proof. Let A ⊂ R2 be the Borel set defining the curve graph Γ. Let κ be aninaccessible cardinal. Let P be a Suslin poset which is placid cofinally belowκ. Let W be the symmetric Solovay model derived from κ, and work in W .Suppose towards contradiction that there is a condition p ∈ P and a P -nameτ such that p forces τ to be a coloring from R2 to ω. We must find two Γ-connected points and a condition stronger than p which forces the two pointsto attain the same color in τ . The name τ and the condition p are definablefrom ground model parameters and an additional parameter z ∈ 2ω. Let V [K]be an intermediate model obtained by a forcing of cardinality smaller than κsuch that z ∈ V [K] and P is placid in V [K]. Work in V [K].

Let p ≤ p be a placid virtual condition. Consider the Cohen poset Q onR2 with its name ygen for the generic point in R2. There must be a cardinalλ ∈ κ, a Q× Coll(ω, λ)-name σ for a condition in the poset P stronger than p,a nonempty open set O ⊂ R2 and a number n ∈ ω such that O Q Coll(ω, λ) Coll(ω,< κ) σ P τ(ygen) = n. Let C be a smooth curve containing nostraight segments, which is a closed subset of R2, such that C ⊂ A and Ocontains some pair of points y0, y1 such that y0 − y1 ∈ C; this is possible as0 is an accumulation point of A. Let X = {〈y0, y1〉 ∈ R2 : y0, y1 ∈ O andy0 − y1 ∈ C}. Let y0, y1 ∈ Y be a pair of points generic over the modelV [K] over the Cohen poset on the space X; so 〈y0, y1〉 ∈ Γ. Theorems 3.2.8and 3.1.4 show that the points y0, y1 ∈ O are separately Q-generic over V [K],and V [K][y0] ∩ V [K][y1] = V [K].

LetH0, H1 ⊂ Coll(ω,< λ) be filters mutually generic over the model V [K][y0, y1],and write p0 = σ/y0, H0 and p1 = σ/y1, H1. By the mutual genericity, the in-


tersection V [K][y0][H0] ∩ V [K][y1][H1] equals to V [K]. By the placidity of thecondition p, the conditions p0, p1 are compatible. By the forcing theorem ap-plied in the respective models V [K][y0][H0] and V [K][y1][H1], in the model Wany common lower bound of the conditions p0, p1 forces in the poset P thatτ(y0) = τ(y1) = n. The proof is complete.

Theorem 12.3.6. Let E be an orbit equivalence on a Polish space resultingfrom a generically turbulent action of some Polish group. In the cofinally placidextensions of the Solovay model, there is no tournament on the E-quotient space.

Proof. The argument depends on a simple abstract property of pairs of inde-pendent continuous open maps on Polish spaces of Definition 3.1.3. Let m ∈ ωbe a number. For each natural number n ∈ m, let Xn be a Polish space, Ynbe Polish spaces and fn : Xn → Yn be continuous open maps. We define theproduct map f =

∏n fn :

∏nXn →

∏n Yn by f(x)(n) = fn(x(n)). It is not

difficult to check that the product map is continuous.

Claim 12.3.7. The set of pairs of independent maps is closed under finiteproduct.

Proof. Let m ∈ ω be a number. For each number n ∈ m let Xn be a Polishspace, Y0n, Y1n be Polish spaces, and let f0n : Xn → Y0n, f1n : Xn → Y1n becontinuous open maps such that for all but finitely many n ∈ ω their rangesare the whole spaces Y0n, Y1n and such that each pair f0n, f1n is independent.Write X =

∏nXn, Y0 =

∏n Y0n and Y1 =

∏n Y1n. We must prove that the

product maps f0 : X → Y0 and f1 : X → Y1 are independent.To this end, let O ⊂ X be a nonempty open set; thinning out the set O if

necessary, we may find open sets On ⊂ Xn for n ∈ m such that O =∏nOn.

Let An ⊂ Y0n be an open set witnessing the independence of the pair f0n, f1n

for the set On. We claim that the open set A =∏nAn ⊂ Y0 witnesses the

independence of the pair f0, f1 for O.To see this, suppose that B0, B1 ⊂ A are nonempty open sets; thinning them

out if necessary, we may find open sets B0n, B1n ⊂ An such that B0 =∏nB0n

and B1 =∏nB1n. For each n ∈ m use the initial assumption on the set An to

find a walk wn of points in On such that for the initial point x0n ∈ wn we havef0n(x0n) ∈ B0n and for the last point x1n ∈ wn we have f0n(x1n) ∈ B1n.

Now, each walk wn is a finite string of points; shifting the indexation, wemay find successive intervals In of natural numbers for n ∈ m such that eachwalk wn is indexed by numbers in In. Let K =

⋃n In and consider the string

of points in the open set O indexed by k ∈ K, defined in the following way: foreach k write n(k) for that n ∈ m for which k ∈ In, and then define point xk ∈ Oby xk(n) = x1n if n < n(k), xk(n) = wk(n) if n(k) = n, and finally xk(n) = x0n

if n(k) < n. It is easy to see that the sequence 〈xk : k ∈ K〉 is the desired walkin the set O from B0 to B1.

Now, towards the proof of Theorem 12.3.6, let Γ be a Polish group acting ona Polish space X in a generically turbulent way, so that E is the resultingorbit equivalence relation. Consider the space Γ × X × X, its closed subset


C = {〈γ, x0, x1〉 : γ · x0 = x1} and the projection maps f0 : C → X into thex0-coordinate and f1 : C → X into the x1 coordinate. The two maps are inde-pendent by Theorem 3.2.2. Let g0, g1 be the product maps f0 × f1 and f1 × f0.Thus, g0, g1 : C × C → X × X are continuous open maps, and they form anindependent pair by the claim.

Let κ be an inaccessible cardinal and let P be a Suslin forcing which is placidcofinally below κ. Let W be the Solovay model derived from κ and work in themodel W . Let p ∈ P be a condition and τ be a P -name such that p τ isa directed graph on the E-quotient space and for each pair 〈c, d〉 of distinctE-equivalence classes, either 〈c, d〉 or 〈d, c〉 is in τ . We must find a conditionq ≤ p and distinct E-classes c, d such that q 〈c, d〉 ∈ τ and 〈d, c〉 ∈ τ .

Both p, τ are definable in the model W from a parameter z ∈ 2ω and pa-rameters in the ground model. Let V [K] be an intermediate forcing extensionobtained by a poset of size < κ containing z and such that P is placid in V [K];work in the model V [K]. By the initial assumption on the poset P , there mustbe a placid virtual condition p ≤ p. Let Q be the Cohen forcing on X × X,adding generic points x0, x1. There must be an open set O ⊂ X × X forcingthe existence of a poset S adding a condition σ ∈ P such that S σ ≤ p andColl(ω,< κ) σ P 〈[x0]E , [x1]E〉 ∈ τ (or the whole statement repeated withx0, x1 interchanged). Since Γ-orbits are dense, it is possible to find open setsO0, O1 ⊂ X and U0, U1 ⊂ Γ such that O0×O1 ⊂ O and (U0 ·O1)×(U1 ·O0) ⊂ O.

Now, consider the Cohen poset R on the space C×C. Consider the conditionU ∈ Q consisting of all tuples 〈γ, x0, x1, δ, x2, x3〉 ∈ C × C such that γ ∈ U0,x0 ∈ O0, δ ∈ U1 and x3 ∈ O1. Let 〈γ, x0, x1, δ, x2, x3〉 be an R-generic tupleover the model V [K]. Note the following:

• 〈x0, x3〉 is a pair Q-generic over V [K], meeting the condition O;

• similarly, 〈x2, x1〉 is a pair Q-generic over V [K], meeting the condition Oby the choice of the neighborhoods U0, U1;

• x0 E x1 and x3 E x2;

• V [K][x0, x3]∩V [K][x2, x1] = V [K] by the fact that g0, g1 are independentmaps and Theorem 3.1.4.

Let S0 = S/x0, x3 and S1 = S/x2, x1. Let H0 ⊂ S0 and H1 ⊂ S1 be filtersmutually generic over the model V [K][xi : i ∈ 4]. By the third item above anda mutual genericity argument, V [K][x0, x3][H0]∩V [K][x2, x1][H1] = V [K]. Letp0 = σ/H0 and p1 = σ/H1. By the placidity of the virtual condition p, theconditions p0, p1 have a lower bound q ∈ P . The forcing theorem finally showsthat in the model W , q both 〈[x0]E , [x2]E〉 and 〈[x2]E , [x0]E〉 belong to τ .

Corollary 12.3.8. Let X be a Borel vector space over a countable field Φ. LetY be an uncountable Polish field and let Ψ be a countable subfield.

1. Let P be the basis poset of Example 6.4.9. In the P -extension of thesymmetric Solovay model, Y does not have a transcendence basis.


2. It is consistent relative to an inaccessible cardinal that ZF+DC holds, Xhas a Hamel basis over Φ, and Y does not have a transcendence basis overΨ.

Corollary 12.3.9. Let E be a Borel equivalence relation on a Polish spaceinduced by a generically turbulent Polish group action. Let F be a Borel equiv-alence relation on a Polish space, classifiable by countable structures.

1. Let P be the linearization poset for F as in Example 8.6.5. In the P -extension of the Solovay model, there is no tournament on the E-quotientspace;

2. it is consistent relative to an inaccessible cardinal that ZF+DC holds, theF -quotient space is linearly ordered, and yet there is no tournament onthe E-quotient space.

12.4 Existence of generic filters

Let P be a Suslin forcing. In the presence of suitable large cardinal axioms,one can replace the P -extension of the symmetric Solovay model with the P -extension of the model L(R). A standard absoluteness argument shows thatvalidity of Π2

1 sentences transfers from the former to the latter, and so thelatter can serve as a vehicle for most independence results presented in thisbook. A natural question arises: does there exist a filter G ⊂ P which is genericover L(R)?

The answer is an easy yes if the poset P in question is σ-closed and theContinuum Hypothesis holds, since then one can meet the dense subsets of theposet P appearing in the model L(R) one by one in a straightforward trans-finite recursion argument. An interesting feature of Bernstein balance is thatit guarantees the existence of generic filters over inner models such as L(R)independently of cardinal arithmetic and assumptions such as the ContinuumHypothesis, purely as a consequence of large cardinal axioms.


1. If B ⊂ P is a set and p ∈ P then write p ≤ B to denote that p is strongerthan all conditions in B;

2. if B ⊂ P then τB is the Coll(ω,B)-name for the analytic set {p ∈ P : p ≤B}.

3. We say that P is soft if for every centered set B ⊂ P of cardinality lessthan c there is a centered set C ⊃ B such that the pair 〈Coll(ω,C), τC〉 isa Bernstein balanced virtual condition.

Note that a soft poset P must have the following centeredness property: ifB ⊂ P is a countable centered set, then B has a lower bound. To see this,

12.4. EXISTENCE OF GENERIC FILTERS 321

just extend it into an Bernstein balanced centered set C ⊂ P , note that in theColl(ω,C)-extension, the set C has a lower bound by the definition, and by aMostowski absoluteness argument the set B must have a lower bound alreadyin the ground model.

Theorem 12.4.2. Suppose that there is a weakly compact Woodin cardinal.Suppose that P is a soft Suslin forcing. Then there is a filter G ⊂ P which isgeneric over the model L(R).

Proof. Let B ⊂ P be a centered set of size < c, and let D ⊂ P be an open denseset in L(R). We will show that there is a condition p ∈ D such that {p} ∪ Bis a centered set. The theorem then follows by a straightforward transfiniterecursion construction, given the fact that there are only c-many subsets of Pin the model of L(R) as soon as a measurable cardinal exists. So, let C ⊃ B bean Bernstein balanced centered set witnessing the definitory properties of thesoft poset P . The set D is definable from a real parameter in the model L(R#).Below, we identify the set D with its definition.

Let κ be a weakly compact Woodin cardinal, let Qκ be the associated count-ably based stationary tower, letK ⊂ Qκ be a generic filter and work in the modelV [K]. Note that κ = ℵ1 and every element of 2ω is generic over V by a posetsmaller than κ. Consider the poset R = Coll(ω,C) and use Proposition 1.7.10 tofind a perfect collection {Hy : y ∈ 2ω} of filters on R pairwise mutually genericover the ground model V . Use Proposition ?? to see the following:

Claim 12.4.3. For every finite set a ⊂ P such that a∪C is a centered set, theset Ya = {y ∈ 2ω : ∃p ∈ V [Hy] p ≤ C and the set {p} ∪ a has no common lowerbound} is countable.

Let j : V → M be the generic ultrapower associated with K; the model Mis closed under ω-sequences. Work in the model M . The closure propertiesof the model M guarantee that the following objects belong to the model M :the collection {Hy : y ∈ 2ω}, the collection {P ∩ V [Hy] : y ∈ Y }, and the mapa 7→ Ya as a varies over finite subsets of P such that a ∪ C is a centered set.Also, the model M satisfies the statement that the range of this map consists ofcountable sets. Looking at the set j(C) in the model M , we see that |j(C)| < c,and for each finite set a ⊂ j(C) the set a ∪ C ⊂ j(C) is centered, and thereforethe set Ya ⊂ 2ω is countable. By a cardinality argument, there must be a pointy ∈ 2ω \

⋃{Ya : a ∈ [j(C)]<ℵ0}.

Work in the model V [Hy]. Note that the theory of L(R#) with real parame-ters is invariant under forcing and so the set D ⊂ P is still open dense in V [Hy].Thus, there must be a condition p ∈ D ∩ V [Hy], p ≤ C. By the choice of thepoint y ∈ Y , the set {p} ∪ j(C) is centered. Thus M |= there is a conditionp ∈ D such that {p} ∪ j(C) is centered. By the elementarity of the embeddingj, M |= there is a condition p ∈ D such that {p} ∪ C is centered; in particular,{p} ∪B is centered as desired.

The following examples all use the Woodin cardinal assumption which is notspelled out.


Example 12.4.4. Let X be a Polish vector space over a countable field. Thereis a basis generic over L(R) for the poset P of Example 6.4.9. This follows fromTheorem 12.4.2 and Theorem ??(1): every centered subset of P yields a linearlyindependent set, which can be extended to a maximal linearly independent set.A maximal linearly independent set yields a Bernstein balanced condition whereE is simply the identity on 2ω.

Example 12.4.5. Let Γ be a Borel graph on a Polish space X. There is amaximal acyclic subgraph ∆ ⊂ Γ generic over L(R) for the poset P of Exam-ple 6.4.10.This follows from Theorem 12.4.2 and Theorem ??(2): every centeredsubset of P yields an acyclic subset of Γ, which can be extended to a maximalacyclic set. A maximal acyclic set yields a Bernstein balanced condition whereE is simply the identity on 2ω.

Example 12.4.6. Let E be a pinned Borel equivalence relation on a Polishspace X. There is a linear ordering on the E-quotient space generic over L(R)for the poset P of Example 8.6.5. This follows from Theorem 12.4.2 and Theo-rem 8.6.4: every centered subset of P yields a partial ordering on the E-quotientspace, which can be extended to a linear order. A linear ordering on the E-quotient space yields a Bernstein balanced condition, by the pinned assumptionon E and Theorem 8.6.4.

Example 12.4.7. Let E,F be pinned Borel equivalence relations on respectivePolish spaces X,Y with uncountably many classes each. There is an injectionfrom the E-quotient to the F -quotient space generic over L(R) for the poset ofDefinition 6.6.2. This follows from Theorem 12.4.2 and Theorem 6.6.3: everycentered subset of P of size < c yields a partial injection from the E-quotientto the F -quotient space and also a set of F -classes which are the forbiddenvalues, both of size < c. A counting argument provides a total injection fromthe E-quotient to the F -quotient space extending the given one and avoidingthe given set of forbidden values. Such an injection yields a Bernstein balancedcondition by Theorem 6.6.3, where G = E × F .

Many other examples can be produced by the observation that the class of softforcings is closed under countable product. The nonexamples are perhaps moreinteresting than the examples.

Example 12.4.8. Let P be the P(ω) modulo finite poset. It is not Bernsteinbalanced by Theorem 12.2.3 above. The P -extension of L(R) contains a Ramseyultrafilter on ω, in particular a P-point. At the same time, it is consistent [82]and occurs in the product Silver model [18] that there are no P-points on ω. Insuch circumstances, there cannot be a filter on P generic over L(R).

Example 12.4.9. Let P be the Lusin poset of Definition 8.8.5. It is not Bern-stein balanced by Theorem 12.2.6 above. The P -extension of L(R) contains aLusin subset of 2ω. At the same time, the negation of Martin’s Axiom and thenegation of the continuum hypothesis implies that every set of size ℵ1 is meager??? and so no Lusin set exists. In such circumstances, there cannot be a filteron P generic over L(R).


Example 12.4.10. Consider the clopen graph Γ on 2ω connecting points x0 6=x1 if the smallest n ∈ ω such that x0(n) 6= x1(n) is even. Let P be the balancedposet of Example 8.8.3. It is not Bernstein balanced by Example 12.2.16 above.In the P -extension of L(R), there is an uncountable set A ⊂ 2ω such that everyΓ-clique and every Γ-anticlique in it is countable. Now, since the graph Γ isclopen, closures of both Γ-cliques and Γ-anticliques are cliques and anticliquesrespectively. Thus, the stated property of A holds also in V . At the sametime, it is consistent that OCA holds in V [91]. In such circumstances, everyuncountable subset of Γ must contain either a Γ-clique or an uncountable Γ-anticlique and therefore a P -generic filter over V cannot exist.

The classes of Bernstein balanced and soft Suslin forcings do not coincide, whichleads to the following question about the existence of generic filters for a par-ticularly simple Bernstein balanced Suslin poset which is not soft.

Question 12.4.11. (In the context of ZFC+suitable large cardinal assump-tion.) Is there a tournament on the F2-quotient space which is generic overL(R) for the poset of Example 8.6.6?

We conclude this section with a mutual genericity result for Bernstein balancedand perfectly balanced forcings.

Theorem 12.4.12. Let κ be a weakly compact Woodin cardinal. Let P0, P1 beσ-closed Suslin forcings such that

1. P0, P1 are ℵ0-tethered;

2. P0 is perfect;

3. every balanced virtual condition in P1 is Bernstein balanced.

If G0 ⊂ P0 and G1 ⊂ P1 are filters (in V ) separately generic over L(R), thenthey are mutually generic over L(R).

Proof. We start with a preliminary observation. Let p0 be the virtual conditionin P0 consisting of a collapse name for the set of all lower bounds of the filterG0; let p1 be defined similarly. By the σ-closure assumption, both p0, p1 arenonzero conditions. Since the filter G0 is generic over L(R), for every analyticset A ⊂ P0 there is a condition in G0 which is either incompatible with allelements of A or is stronger than some element of A. By the ℵ0-tether of theposet P0, it must then be the case that p0 is a balanced virtual condition in P0.By the same reasoning, p1 is a balanced virtual condition in P1, and it is evenBernstein balanced by the assumption (3).

Now, let X be any uncountable Polish space. In view of Proposition 1.7.8applied in the model L(R), it is enough to show that for any two disjoint setsA0, A1 ⊂ X in the respective models L(R)[G0], L(R)[G1], there is a set B ⊂ Xin L(R) containing A1 and disjoint from A0. To this end, let τ0, τ1 be respectiveP0, P1-names in L(R) such that A0 = τ0/G0 and A1 = τ1. The names τ0, τ1 canbe viewed as sets of reals, and so each has a definition in L(R) from real and


ordinal parameters; below, we will evaluate them in other models using thesedefinitions. We also evaluate P0, P1 using their definitions. Write 0 for theforcing relation of P0 and 1 for the forcing relation of P1.

Let H ⊂ Qκ be a generic filter. Let j : V → M be the associated genericultrapower and work in the model M . Note that G1 ⊂ j(G1) is a countable setin the model M and by the σ-closure assumption on the poset P1, the filter j(G1)must contain a lower bound of G1. Consider the set B = {x ∈ X : ∃p ∈ P1 p isa lower bound of the filter G1 and p 1 x ∈ τ1}. Clearly, the set B ⊂ X belongsto L(R) and it contains the set j(A1). By the elementarity of the embedding j,it will be enough to show that B is disjoint from the set j(A0).

To see this, we prove the following claim, which is also a statement of themodel M . Call a Borel set C ⊂ X positive if for every condition p1 ∈ P1 whichis a lower bound of the filter G1, p1 1 C ∩ τ1 6= 0. By the forcing theoremapplied to j(G1), C ∩ A1 6= 0 holds for every positive Borel set C. Clearly, thecollection of positive Borel sets is in L(R).

Claim 12.4.13. If p0 ∈ P0 is a condition which is a lower bound of the filter G0

and x ∈ B is a point such that p0 0 x ∈ τ0, then there is a condition p′0 ≤ p0

and a positive Borel set C ⊂ X such that p′0 0 C ⊂ τ0.

The theorem immediately follows. By the claim and the genericity of the filterj(G0) over L(R), there must be in j(G0) either a condition which forces B∩τ0 =0 or a condition which forces C ⊂ τ0 for some Borel positive set C ⊂ X.However, the latter alternative means that j(A0)∩ j(A1) 6= 0, contradicting ourinitial assumptions on A0, A1. The former alternative confirms that the set Bseparates j(A0) and j(A1) as required.

To prove the claim, fix p0 ∈ P0 and x ∈ X as in the assumption. In view ofthe fact that the reals of the model M are the reals of some symmetric Solovaymodel derived from κ (Fact 1.7.18(3)), in the ground model there must be aposet Q of size < κ and Q-names σ0, σ1, η for conditions in P0, P1 and a pointin X respectively such that Q forces that σ0 is a lower bound of G0, σ1 is a lowerbound of G1 and Coll(ω,< κ) σ0 0 η ∈ τ0 and Coll(ω,< κ) σ1 1 η ∈ τ1.Moreover, there has to be a filter K ⊂ Q generic over V such that p0 = σ0/Kand x = η/K. Now, let H ⊂ Coll(ω,P(Q)∩V ) be a filter in M mutually genericover V with the filter K.

Work in the model V [H]. Use Proposition 1.7.10 to find a continuous func-tion y 7→ Hy which to each element y ∈ 2ω assigns a filter Hy ⊂ Q so that forevery finite set a ⊂ 2ω, the filters {Hy : y ∈ a} are mutually generic over V . Forevery element y ∈ 2ω, write p0y = σ0/Ky, p1y = σ1/Ky, and xy = η/Ky. Sincethe conditions p0y ∈ P0 are all stronger than the balanced virtual condition p0

and they are found in mutually generic models, each finite set of them has alower bound as in Claim 12.1.4. By the perfectness assumption on the posetP0, the model V [H] contains a perfect set D ⊂ 2ω and a condition r ∈ P0

which is stronger than all p0y for all y ∈ D. Since both conditions p0, r ∈ P0

are stronger than the balanced virtual condition p0 and are found in mutuallygeneric models, they are compatible with a lower bound p′0.


Now, consider the set {p1y : y ∈ D} and the perfect set C = {xy : y ∈ D}. Bythe Bernstein balance of the virtual condition p1, every condition p1 ∈ P1 whichis a lower bound of the filter G1 is compatible with p1y for all but countablymany points y ∈ D. As a result, C ⊂ X is a positive Borel set. The initialchoice of the name η also implies that p′0 C ⊂ τ0. The claim has just beenproved.


Chapter 13

The arity divide

13.1 m,n-centered and balanced forcings

In this chapter, we develop a tool which is particularly useful for ruling outcombinatorial objects with high degree of organization. The following notion iscentral:

Definition 13.1.1. Let P be a Suslin poset and n ∈ m be numbers.

1. A virtual condition p in P is m,n-balanced if in every generic extension,whenever {Hi : i ∈ m} are filters generic over V such that for every seta ⊂ m of size n the filters {Hi : i ∈ a} are mutually generic over V , ifpi ≤ p are conditions in P in V [Hi], then the set {pi : i ∈ m} has acommon lower bound in P .

2. The poset P is m,n-balanced if for every condition p ∈ P there is a m,n-balanced virtual condition p ≤ p.

As an initial example, let P be the poset of linear orderings on countable setsof E0-classes, ordered by reverse extension. The balanced virtual conditions areclassified by linear orderings of the E0-quotient space. Each such ordering p isin fact m, 2-balanced for every m ∈ ω. To see this, in some ambient forcingextension let V [Gi] for i ∈ m be generic extensions which are pairwise mutuallygeneric, and let pi ∈ V [Gi] be a condition stronger than p for each i ∈ m.The domains of the linear orderings pi pairwise intersect in the set of groundmodel E0-classes; on this set, the orderings pi agree and yield the ordering p.Therefore, it is possible to find a common linearization of

⋃i pi, which is their

common lower bound.Sometimes, a much simpler notion makes an appearance:

Definition 13.1.2. Let n ∈ m be numbers. A poset P is m,n-centered ifwhenever a ⊂ P is a set of size m such that any subset of a of size n has acommon lower bound, then a has a common lower bound.

327

328 CHAPTER 13. THE ARITY DIVIDE

As an initial example, let P be the partial order of all tournaments on count-able sets of E0-classes, ordered by reverse inclusion. In it, every countable set ofpairwise compatible conditions has a common lower bound. It is immediatelyclear that every balanced poset which is m,n-centered is also m,n-balancedand each balanced virtual condition in it is m,n-balanced. An important partof this chapter is devoted to detecting distinctions between m,n-balanced andm,n-centered forcings. Here, we restrict ourselves to two initial observations.Note that m,n-centeredness (unlike the corresponding balance notion) is Π1

2

and therefore absolute between all generic extensions. Note also that m,n-centeredness is a combinatorial (as opposed to forcing) property of the theposet, in the sense that one can have two analytic dense subsets Q0, Q1 of agiven Suslin poset P such that Q0 is m,n-centered while Q1 is not. On theother hand, m,n-balance is decidedly a forcing property of posets.

As a final remark, observe that both m,n-balance and m,n-centerednessare properties preserved by countable support product. Thus, the numerousindependence results in this chapter can be combined.

13.2 Preservation theorems

The main reason for considering the m,n-balance is that it rules out discon-tinuous homomorphisms between Polish groups in balanced extensions of thesymmetric Solovay model. One way of doing that is to prove that in the givenmodel of ZF+DC, the chromatic number of the Hamming graph H2 is not two(say using Theorem 11.4.13 or 14.4.7), and then use a result of [76] to show thatin ZF+DC this rules out discontinuous homomorphisms. The following theo-rem works in many cases which are not treatable using the chromatic numberof the Hamming graph. Note that a nonprincipal ultrafilter on ω yields in ZFa discontinuous homomorphism from 2ω to 2; thus, ruling out discontinuoushomomorphisms automatically rules out nonprincipal ultrafilters on ω as well.The following theorems are stated using Convention 1.7.16.

Theorem 13.2.1. Let n ∈ ω be a number. In cofinally n + 1, n-balanced ex-tensions of the symmetric Solovay model, all homomorphisms between Polishgroups are continuous.

Proof. Suppose that κ is an inaccessible cardinal. Suppose that P is a Suslinforcing which is is n + 1, n-balanced cofinally below κ. Let W be a symmetricSolovay model derived from κ and work in W . Suppose towards a contradictionthat the conclusion of the theorem fails: thus, there are Polish groups Γ,∆, acondition p ∈ P and a P -name τ such that p τ : Γ → ∆ is a discontinuoushomomorphism. Both p and τ must be definable in W from parameters inthe ground model and some point z ∈ 2ω. Let V [K] be some intermediateextension of V by a poset of cardinality smaller than κ such that z ∈ V [K] andP is n+ 1, n-balanced in V [K]. Work in the model V [K].

Let p ≤ p be a n+ 1, n-balanced condition. Consider the poset PΓ adding ageneric element γgen of the group Γ. The following is a key claim:

13.2. PRESERVATION THEOREMS 329

Claim 13.2.2. PΓ forces that there are disjoint basic open sets O0, O1 ⊂ ∆ suchthat Coll(ω,< κ) ∃p0 ≤ p p0 P τ(γgen) ∈ O0 and ∃p1 ≤ p p1 τ(γgen) ∈ O1.

Proof. Suppose towards a contradiction that q ∈ PΓ forces the opposite. Then,q forces that in the Coll(ω,< κ)-extension there is a unique point δ ∈ ∆ suchthat there is a condition p′ ≤ p in the poset P forcing τ(γgen) = δ. Since thepoint δ is in the symmetric Solovay model definable from τ, p, and γgen , it must

belong to the PΓ-extension, and there is a PΓ-name δ for it. Note that then,q Coll(ω,< κ) p P δ = τ(γgen).

Now, pass to the symmetric Solovay model W and consider the set B = {γ ∈Γ: γ is PΓ-generic over V [K] and γ ∈ q} and the function f : B → ∆ assigningto each point γ ∈ B the point δ/γ. The function f is continuous on B, and thechoice of the name δ implies that p f ⊂ τ holds. The set B ⊂ Γ is dense Gδin q. A standard result in the theory of Polish groups [55, Theorems 9.9 and9.10] now says that a homomorphism from Γ to ∆ which is continuous on a setcomeager in a nonempty open set is in fact continuous on the whole group Γ.This contradicts the initial choice of the name τ .

Let X = {x ∈ Γn+1 :∏i∈n x(i) = x(n)}; this is a closed subset of Γn+1,

equipped with the topology inherited from Γn+1. If a ⊂ n + 1 is a set of sizen, it is clear that the remaining coordinate of a point x ∈ X is a continuousfunction of the coordinates x(i) for i ∈ a, and therefore the projection from Xto Γa is a continuous and open surjection. Let x ∈ X be a point PX genericover V [K]. Use Proposition 3.1.1 to see that whenever a ⊂ n + 1 are distinctnumbers then x � a ∈ Γ is a sequence (PΓ)n-generic over V [K].

Fix i ∈ n. Working in the model V [K][x(i)] find a poset Ri of size < κand a name σi for a condition in the poset P stronger than p and a name δifor an element of the group ∆ such that V [K][x(i)] |= Ri Coll(ω,< κ) σi P τ(x(i)) = δi. In the model V [K][x(n)], find the disjoint basic open setsO0, O1 ⊂ ∆ as in Claim 13.2.2. Passing to a condition in the product

∏i∈nRi

and switching O0, O1 if necessary, we may assume that V [K][x � n] |=∏i∈nRi ∏

i∈n x(i) /∈ O0. Use the choice of the set O0 to find, in the model V [K][x(n)], a

poset Rn of size < κ, Rn-names σn for an element of P stronger than p and δnfor an element of O0 ⊂ ∆ such that Rn Coll(ω,< κ) σn P τ(x(n)) = δn.

Let Hi ⊂ Ri for i ∈ n+1 be filters mutually generic over the model V [K][x].For each i ∈ n + 1 write pi = σi/Hi ∈ P and δi = δi/Hi. Note that whenevera ⊂ n is a set of size n, the models V [K][xi][Hi] for i ∈ a are mutually genericextensions of the model V [K]. By the balance assumption on the virtual condi-tion p, the conditions pi ∈ P for i ∈ n+1 have a lower bound, call it q ∈ P . LetW be a symmetric Solovay extension of the model V [K][x][Hi : i ∈ n + 1] andwork in W . The condition q forces that τ(x(i)) = δi for all i ∈ n+ 1. Observethat

∏i∈n = x(n) and

∏i∈n δi 6= δn since

∏i∈n δi /∈ O0 while δn ∈ O0. This

contradicts the assumption that τ is forced to be a homomorphism from Γ to∆.

Theorem 13.2.1 has a counterpart for certain quotient groups. Suppose that I


is a Borel ideal on ω. The quotient space P(ω)/I will be viewed as a groupwith the (quotient of the) symmetric difference operation. A homomorphismh : P(ω)/I → P(ω)/J is Borel if the set {〈x, y〉 ∈ P(ω)×P(ω) : h([x]I) = [y]J}is Borel.

Theorem 13.2.3. Let I, J be Borel ideals on ω containing all singletons andsuppose that the modulo J equality is a pinned equivalence relation. Let n ∈ ωbe a number. In cofinally n+ 1, n-balanced extensions of the symmetric Solovaymodel, all homomorphisms of P(ω)/I to P(ω)/J are Borel.

The assumptions are satisfied in particular for the Fσ-ideals J (as then themodulo J equality is an Fσ-equivalence relation and as such pinned [48, Theorem17.1.3 (iii)]) and for the analytic P-ideals J (as they are Borel and Polishableby [85]. The modulo J equality is an orbit equivalence relation of the action ofthe Polish abelian group J by the symmetric difference, and orbit equivalencerelations of abelian Polish groups are pinned by [48, Theorem 17.1.3 (ii)]).

Proof. Write E for the modulo I equality and F for modulo J equality, bothBorel equivalence relations on X = P(ω). Suppose that κ is an inaccessiblecardinal. Suppose that P is a Suslin forcing which is is n+1, n-balanced cofinallybelow κ. Let W be a symmetric Solovay model derived from κ and work inW . Suppose towards a contradiction that the conclusion of the theorem fails.Thus, there is a condition p ∈ P and a P -name τ such that p forces τ is ahomomorphism. Both p and τ must be definable in W from parameters in theground model and some point z ∈ 2ω. Let V [K] be a generic extension of V by aposet of cardinality smaller than κ such that z ∈ V [K] and P is n+1, n-balancedin V [K].

Work in the model V [K]. Let p ≤ p be a n + 1, n-balanced condition.Consider the Cohen poset Q on X adding a generic set xgen ⊂ ω. There aretwo cases.

Case 1. Q ∀y ⊂ ω Coll(ω,< κ) ∃r ≤ p r P τ([xgen ]E) 6= [y]F holds.This case in fact leads to a contradiction. Write ∆ for the symmetric differenceoperation on X and utilize its associativity to apply it also to finite tuples ofsubsets of ω. Let X = {x ∈ Xn+1 : ∆i∈nx(i) = x(n)}; this is a closed subset ofXn+1, equipped with the topology inherited from Xn+1. If a ⊂ n+ 1 is a set ofsize n, it is clear that the remaining coordinate of a point x ∈ X is a continuousfunction of the coordinates x(i) for i ∈ a, and therefore the projection from Xto Γa is a continuous and open surjection. Let x ∈ X be a point PX genericover V [K]. Use Proposition 3.1.1 to see that whenever a ⊂ n + 1 are distinctnumbers then x � a ∈ Γ is a sequence Qn-generic over V [K].

Fix i ∈ n. Working in the model V [K][x(i)] find a poset Ri of cardinalitysmaller than κ and an Ri-name σi for a condition in the poset P stronger thanp and an Ri-name ηi for a such that V [K][x(i)] |= Ri Coll(ω,< κ) σi Pτ([x(i)]E) = [ηi]F . Let Hi ⊂ Ri for i ∈ n be filters mutually generic over themodel V [K][x]. Write pi = σi/Hi and yi = ηi/Hi for i ∈ n, and look at the sety = ∆i∈nyi ⊂ ω. There are two hopeless subcases.


Case 1a. The equivalence class [y]F is represented in the model V [K][x(n)].Work in the model V [K][x(n)]. Use the initial case assumption to find a posetRn, an Rn-name σn for a condition in P such that Rn Coll(ω,< κ) σn τ([x(n)]E) 6= [y]F . Let Hn ⊂ Rn be a filter generic over the modelV [K][x][Hi : i ∈ n] and let pn = σn/Hn. Note that for every j ∈ n + 1, themodels V [K][x(i)][Hi] for i ∈ n + 1 \ {j} are mutually generic over V [K]. Itfollows from the n + 1, n-balance assumption on the condition p that the con-ditions pi for i ∈ n+ 1 have a common lower bound, say r ∈ P . The conditionr forces τ([x(i)]E) = [yi]F for all i ∈ n, and τ(xn) is not modulo J equiva-lent to y = ∆i∈nyi. This contradicts the assumption that τ is forced to be ahomomorphism, as xn = ∆i∈nxi.Case 1b. Case 1a fails. Work in the model V [K][x(n)]. Find a poset Rn, anRn-name σn for a condition in P and an Rn-name ηn for a subset of ω suchthat Rn Coll(ω,< κ) σn τ([x(n)]E) 6= [ηn]F . Let Hn ⊂ Rn be a filtergeneric over the model V [K][x][Hi : i ∈ n] and let pn = σn/Hn and yn = ηn/Hn.Now, either the name ηn is F -pinned below some condition in the filter Hn, inwhich case the F -class [yn]F is represented in V [K][x(n)] by the assumptionthat F is pinned, or the name ηn is not F -pinned below any condition in thefilter Hn, in which case the F -class [yn]F is not represented even in the modelV [K][x(i) : i ∈ n][Hi : i ∈ n] by the mutual genericity. In either case, y F ynfails.

The rest of Case 1b is identical to Case 1a. For every index j ∈ n + 1,the models V [K][x(i)][Hi] for i ∈ n + 1 \ {j} are mutually generic over V [K].It follows from the n + 1, n-balance assumption on the condition p that theconditions pi for i ∈ n + 1 have a common lower bound, say r ∈ P . Thecondition r forces τ([x(i)]E) = [yi]F for all i ∈ n, and τ(xn) = yn is not moduloJ equivalent to y = ∆i∈nyi. This contradicts the assumption that τ is forcedto be a homomorphism, as xn = ∆i∈nxi.Case 2. Case 1 fails. By the Borel reading of names for the poset Q, we can finda Borel set B ⊂ X co-meager in some nonempty open set O ⊂ X and a Borelfunction f : B → X such that O Coll(ω,< κ) p τ([xgen ]E) = [f(xgen)]F .Now, in the model W let C ⊂ B be the set of all points which are Q-genericover V [K]. Note that the set C ⊂ B is comeager in O and Borel, and by theforcing theorem, p P ∀x ∈ C τ([x]E) = [f(x)]E . Since the set C is Boreland non-meager, for all infinite sets y ∈ X there are sets x0, x1 ∈ C such thatx0∆x1 = y modulo finite (Pettis theorem, [55, Theorem 9.9]). It is then clearthat p forces the homomorphism τ to be Borel: τ([y]E) = [z]F just in case thereexist x0, x1 ∈ C such that x0∆x1 = y modulo finite and z F f(x0)∆f(x1),and τ([y]E) = [z]F just in case for all x0, x1 ∈ C such that x0∆x1 = y modulofinite, z F f(x0)∆f(x1). These are in turn analytic and coanalytic descriptionsof τ .

The various degrees of n+1, n-balance are not difficult to separate. One partic-ularly elegant way of doing so is to seek monochromatic solutions to equationsin Polish groups. We state one prominent case, leaving the others to the patientreader.


Theorem 13.2.4. Let 〈Γ, ·〉 be an uncountable Polish group such that the mapγ 7→ γ · γ is a self-homeomorphism of Γ. In cofinally 3, 2-balanced extensions ofthe symmetric Solovay model, for every coloring c : Γ→ ω there is a solution tothe equation xy = zz consisting of monochromatic, pairwise distinct points inΓ.

Proof. Let κ be an inaccessible cardinal. Let P be a Suslin poset which iscofinally 3, 2-balanced below κ. Let W be the symmetric Solovay model derivedfrom κ and work in W . Let τ be a P -name and p ∈ P be a condition forcingτ to be a map from Γ to ω. We must find a solution to the equation xx = yzconsisting of pairwise distinct points and a condition stronger than p whichforces the point of the solution to have all the same color.

The name τ and the condition p must both be definable with parameters inthe ground model and another parameters z ∈ 2ω. Let V [K] be an intermediatemodel obtained by a poset of cardinality smaller than κ such that z ∈ V [K]and P is 3, 2-balanced in V [K]. Work in the model V [K]. Let p ≤ p be a3, 2-balanced virtual condition. Let PΓ be the Cohen poset on the group Γ withits name for the generic point γgen in Γ. There must be a cardinal λ < κ, anumber n ∈ ω, a PΓ × Coll(ω, λ)-name σ and a condition O ∈ PΓ such thatO Coll(ω, λ) Coll(ω,< κ) σ P τ(γgen) = n.

Let γ ∈ O be an arbitrary point, and let δ ∈ Γ be a point so close (but notequal) to the unit that both γδ and δ−1γ belong to the open set O ⊂ Γ. Notethat the points γδ, δ−1γ, and γ are pairwise distinct and solve the equationxy = zz. (If γδ = δ−1γ then the square of this point would give the sameresult as γγ, contradicting the assumption that the squaring function is a self-homeomorphism, in particular a bijective self-map, of Γ.) Let O0, O1, O2 ⊂ Obe pairwise disjoint open sets separating these three points, and let X = {x ∈O0 × O1 × O2 : x(0)x(1) = x(2)x(2)}. This is a relatively closed subset ofO0 × O1 × O2. Moreover, each coordinate of a point in X is a continuousfunction of the other two (in the case of the third coordinate this again usesthe assumption that the squaring function is a self-homeomorphism of Γ) andtherefore the projection maps from X into any pair of coordinates are openmaps from X to Γ2.

Consider the Cohen poset PX . Let x ∈ X be a triple of points which isPX -generic over the model V [K]. Use Proposition 3.1.1 to see that these pointsare pairwise mutually PΓ-generic below the condition O. Let Hi ⊂ Coll(ω, λ)for i ∈ 3 be filters mutually generic over the model V [X][x], and let pi = σi/Hi.Since the extensions V [K][x(i)][Hi] for i ∈ 3 are pairwise mutually generic, thebalance assumption on the virtual condition p shows that the conditions pi fori ∈ 3 have a common lower bound, say r ∈ P . By the forcing theorem appliedin the respective models V [K][x(i)][Hi] for i ∈ 3, we see that r forces all threepoints in the triple x to have the same color n. At the same time, the triplesolves the equation xy = zz as required.

Finally, we move to the much better organized world of 3, 2-centered forcings.


Theorem 13.2.5. Let Γ be an acyclic Borel graph on a Polish space X suchthat the Γ-path connectedness relation is Borel and unpinned. In 3, 2-centered,cofinally balanced extensions of the symmetric Solovay model, Γ has no maximalacyclic subgraph.

Proof. Let κ be an inaccessible cardinal. Let P be a Suslin, 3, 2-centered posetwhich is balanced cofinally below κ. Let W be a symmetric Solovay modelderived from κ. Work in W . Suppose that p ∈ P is a poset and τ is a P -namesuch that p forces τ to be a spanning subset of Γ. We will find a cycle in Γ anda condition stronger than p which forces the cycle to be a subset of τ .

The condition p and the name τ are definable from ground model parametersand an additional parameter in 2ω. Let V [K] be an intermediate extensioncontaining z obtained by a poset of cardinality smaller than κ and such thatP is balanced in V [K]. Work in the model V [K]. Let p ≤ p be a balancedvirtual condition in P . The equivalence relation E is not pinned in V [K] byCorollary 2.7.3, and there is a nontrivial E-pinned name on the poset collapsingℵ1 to ℵ0 by Theorem 2.6.3. By the cofinal balance assumption, there mustbe a poset Q of cardinality smaller than κ, a nontrivial E-pinned Q-name ηfor an element of X, and a Q-name σ for a balanced virtual condition in theQ-extension which is stronger than p. Let H0, H1, H2 ⊂ Q be filters mutuallygeneric over V [K] and write pi = σ/Hi and xi = η/Hi for i ∈ 3. For distinctindices i, j ∈ 3, we conclude by the balance of the condition p in V [K] that thevirtual conditions pi, pj must be compatible. Thus, in the model V [K][Hi][Hj ],there must be a poset Qij and Qij-name σij for a condition in P stronger thanboth pi and pj , and a Qij-name χij for a Γ-path between xi and xj such thatV [K][Hi][Hj ] |= Qij Coll(ω,< κ) σij χij ⊂ τ .

Let Gij ⊂ Qij be filters mutually generic over the model V [K][Hi : i ∈ 3].Write pij = σij/Gij and uij = χij/Gij . Observe that whenever i ∈ 3 and j0, j1 ∈3 are the two indices distinct from i, then pij0 and pij1 are compatible conditionsin P by the balance of the condition pi in V [K][Hi]. By the 3, 2-centerednessassumption, the conditions pij for i, j ∈ 3 distinct have a common lower boundr ∈ P . Note also that after erasing some local loops if necessary

⋃ij uij is

a cycle. By the forcing theorem applied in each model V [K][Hi, Hj ][Gij ], itfollows that r

⋃ij uij ⊂ τ as desired.

Theorem 13.2.6. Let X be an uncountable Polish space. In 3, 2-centered,cofinally balanced extensions of the symmetric Solovay model, every set mappingf : [X]2 → Xℵ0 has a free triple.

Proof. Let κ be an inaccessible cardinal. Let P be a Suslin, 3, 2-centered posetwhich is balanced cofinally below κ. Let W be a symmetric Solovay modelderived from κ. Work in W . Suppose that p ∈ P is a poset and τ is a P -namesuch that p forces τ to be a function from [X]2 to [X]ℵ0 . We will find a tripleof points in X and a condition stronger than p which forces the triple to be freefor τ .

The condition p and the name τ are definable from ground model parametersand an additional parameter in 2ω. Let V [K] be an intermediate extension


containing z obtained by a poset of cardinality smaller than κ and such thatP is balanced in V [K]. Work in the model V [K]. Let p ≤ p be a balancedvirtual condition in P . By the cofinal balance assumption, there must be aposet Q of cardinality smaller than κ, a Q-name η for an element of X whichdoes not belong to V [K], and a Q-name σ for a balanced virtual conditionin the Q-extension which is stronger than p. Let H0, H1, H2 ⊂ Q be filtersmutually generic over V [K] and write pi = σ/Hi and xi = η/Hi for i ∈ 3.For distinct indices i, j ∈ 3, we conclude by the balance of the condition p inV [K] that the virtual conditions pi, pj must be compatible. Thus, in the modelV [K][Hi][Hj ], there must be a poset Qij and Qij-name σij for a condition in Pstronger than both pi and pj , and a Qij-name χij for an element of Xω suchthat V [K][Hi][Hj ] |= Qij Coll(ω,< κ) σij rng(χij) = τ(xi, xj).

Let Gij ⊂ Qij be filters mutually generic over the model V [K][Hi : i ∈ 3].Write pij = σij/Gij and uij = χij/Gij . Observe that whenever i ∈ 3 andj0, j1 ∈ 3 are the two indices distinct from i, then pij0 and pij1 are compatibleconditions in P by the balance of the condition pi in V [K][Hi]. By the 3, 2-centeredness assumption, the conditions pij for i, j ∈ 3 distinct have a commonlower bound r ∈ P . Note also that for each k ∈ 3, the point xk does not belongto the model V [K][Hi, Hj ][Gij ] where i, j ∈ 3 are the two indices distinct from k,and therefore does not belong to the range of uij . It follows that r {xi : i ∈ 3}is a free set for τ as desired.

Theorem 13.2.7. Let E be a pinned Borel equivalence relation on a Polishspace X. In 3, 2-centered, tethered, and cofinally balanced extensions of thesymmetric Solovay model, the following are equivalent:

1. E has a transversal;

2. the E-quotient space is linearly ordered.

Proof. The implication (1)→(2) is trivial. In ZF, the space X, as every Polishspace, carries a Borel linear ordering ≤. If A ⊂ X is an E-transversal, then onecan order the E-quotient space by setting c ≺ d if the unique element of A ∩ cis ≤-smaller than the unique element of A ∩ d. The implication (2)→(1) is theheart of the matter.

Let κ be an inaccessible cardinal. Let P be a Suslin forcing which is 3, 2-centered, balanced cofinally below κ and tethered below κ. Let W be a sym-metric Solovay model derived from κ and work in W . Suppose that p ∈ P is acondition forcing E not to have a transversal. Suppose that τ is a P -name for atournament on the E-quotient space. We will find a 3-cycle on the E-quotientspace and a condition stronger than p which forces the cycle to be a subset ofτ . This will prove the theorem.

The condition p and the name τ are definable from some ground modelparameters and an additional parameter z ∈ 2ω. As a provisional definition,call a tuple 〈M, p,Q, η, σ〉 symmetric if M is an intermediate generic extensionof V by a poset of cardinality smaller than κ such that z ∈ M , p ≤ p is abalanced virtual condition in the model M , Q ∈ M is a poset of cardinality


smaller than κ, η ∈M is a Q-name for an element of X which is not E-relatedto any element of M ∩X, σ ∈M is a Q-name for a balanced virtual conditionin the Q-extension of M which is stronger than p, and

(*) for any conditions q0, q1 ∈ Q, in some generic extension there are filtersH0, H1 ⊂ Q separately generic over M such that q0 ∈ H0, q1 ∈ H1, η/H0

is E-related to η/H1, and σ/H0 is compatible with σ/H1 in the poset P .

The following claim is central.

Claim 13.2.8. There is a symmetric tuple 〈M, p,Q, η, σ〉.

Proof. LetG ⊂ P be a filter generic over the modelW containing the condition pand work in W [G]. For each E-class c, consider the model Mc of sets hereditarilydefinable from parameters in the ground model and the additional parametersz, c,G. There must be an E-class c such that c ∩Mc = 0; otherwise, one couldform an E-selector in W [G] as the set A = {x ∈ X : x is the least element of[x]E in the canonical well-order of M[x]E}, contradicting the initial assumptionson the condition p. Fix such an E-class c, an arbitrary element x ∈ c, and letMx be the model of sets hereditarily definable from parameters in the groundmodel and the additional parameters z, x,G. Note that Mc ⊂Mx holds.

By the balance assumption, the models Mc and Mx are generic extensions ofthe ground model by posets of cardinality smaller than κ; in particular, Mx is ageneric extension of Mc by a poset of cardinality smaller than κ. By the tetherassumption, as in Proposition ???, there are virtual balanced conditions pc andpx in the respective models Mc and Mx such that their realizations belong tothe generic filter G. Necessarily px ≤ pc ≤ p holds. Working in Mc, let Q bea poset of cardinality smaller than κ such that there is a filter H ⊂ Q genericover Mc such that Mx = Mc[H]. Let η ∈ Mc be a Q-name such that x = η/Hand let σ be a Q-name such that px = σ/H. We claim that for some conditionq ∈ Q, the tuple 〈M, pc, Q � q, η, σ〉 is symmetric.

To see this, in W [G] let D ⊂ Q be the set of all conditions r ∈ Q suchthat there is a filter H ′ ⊂ Q which is generic over Mc and such that η/H ′ ∈ cand σ/H ′ has a realization in the filter G. The set, having just been definedfrom c and G, belongs to the model Mc. It also contains the Q-generic filterH ⊂ Q as a subset. By a density argument with the filter H, there must be acondition q ∈ H such that D contains all conditions r ∈ Q such that r ≤ q. Itis immediate from the definition of the set D that the condition q ∈ Q works asrequired.

Now, fix the symmetric tuple 〈M, p,Q, η, σ〉. Let H0 ⊂ Q0 and H1 ⊂ Q1 befilters mutually generic over M , and write x0 = η/H0, x1 = η/H1, p0 = σ/H0

and p1 = σ/H1.

Claim 13.2.9. In some generic extension N of M [H0, H1] by a poset of cardi-nality smaller than κ there is a condition r ∈ P stronger than p0, p1 such thatN |= Coll(ω,< κ) r P 〈[x0]E , [x1]E〉 ∈ τ .


Proof. To start with, note that Q × Q forces σleft and σright to be compatiblevirtual conditions in P by the balance assumption on p, and it also forces ηleft

and ηright to be E-unrelated elements of X by the pinned assumption on theequivalence relation E. Let 〈q0, q1〉 ∈ H0, q1 ∈ H1 be conditions and supposetowards a contradiction that in M , the condition 〈q0, q1〉 forces the negation ofthe statement of the claim.

Let λ < κ be a cardinal larger than |P(Q)∩M | and let K0,K1 ⊂ Coll(ω, λ)be filters mutually generic over M [H0, H1]. By (*) and the forcing theorem,there are filters H ′0 ∈ M [H0][K0] and H ′1 ∈ M [H1][K1] which are generic overM , q0 ∈ H ′1 and q1 ∈ H ′0 holds, and moreover, writing x′0 = η/H ′0, x′1 = η/H ′1,p′0 = σ/H ′0 and p′1 = σ/H ′1, we have that x0 E x′0 and x1 E x′1 both hold, andp0 and p′0 are compatible virtual conditions in P , and p1 and p′1 are compatiblevirtual conditions in P . Note also that the filters H ′0, H

′1 ⊂ Q are mutually

generic overM since they are found in the respective mutually generic extensionsM [H0][K0] and M [H1][K1]–Corollary 1.7.9.

Now, observe that the conditions p0, p′0, p1, p

′1 have a common lower bound

in the poset P . To see this, note that p0, p′0 are compatible with a lower bound

r0 ∈ P in the model M [H0][K0], and p1, p′1 are compatible with a lower bound

r1 ∈ P1 in the model M [H1][K1]. Now use the balance of the condition p to seethat r0, r1 are compatible conditions in P .

Finally, there must be a generic extension N of the model M [H0, H1][K0,K1]by a poset of cardinality smaller than κ in which there is a condition r ∈ Pwhich is a common lower bound of r0, r1 and such that N |= Coll(ω,< κ) r P 〈[x0]E , [x1]E〉 ∈ τ or N |= Coll(ω,< κ) r P 〈[x0]E , [x1]E〉 ∈ τ .The former option violates the initial contradictory assumption in view of theforcing theorem in the model M and the generic filters H0, H1, and the latteroption violates the initial contradictory assumption in view of the generic filtersH ′0, H

′1.

Finally, let H0, H1, H2 ⊂ Q be filters mutually generic over the model M .For each i ∈ 3 write xi = η/Hi and pi for σ/Hi. Use the claim to find a posetR01 ∈M [H0][H1] of cardinality smaller than κ and an R-name χ01 for a commonlower bound of p0, p1 in P such that M [H0][H1] |= R01 Coll(ω,< κ) χ01 〈[x0]E , [x1]E〉 ∈ τ . Similarly for R12, χ12 and R20, χ20. Let K01 ⊂ R01, K12 ⊂R12, and K20 ⊂ R20 be filters mutually generic over the model M [Hi : i ∈ 3].Write p01 = χ01/K01 and similarly for p12, p20. Note that the conditions p01

and p12 are compatible in P by the balance of the virtual condition p1 ∈M [H1]:they are found in the respective extensions M [H1][H0][K01] and M [H1][H2][K12]mutually generic over M [H1], and they are both stronger than p1. Similarly,the conditions p12 and p20 are compatible, and the conditions p20 and p01 arecompatible. Finally, the 3, 2-centeredness assumption shows that the conditionsp01, p12, p20 have a common lower bound q ≤ p. The forcing theorem appliedin the respective models M [H0, H1][K01], M [H1][H2][K12] and M [H2][H0][K20]shows that in the model W , q forces the sequence 〈[x0]E , [x1]E , [x2]E , [x0]E〉 toform an oriented 3-cycle which is a subset of τ . The proof of the theorem iscomplete.

13.3. EXAMPLES 337

13.3 Examples

The first batch of examples deals with the implications and limitations of The-orem 13.2.7. We use Theorem 13.2.1 to add the nonexistence of discontinuoushomomorphisms of Polish groups to the conclusions for reference purposes.

Example 13.3.1. Let E be a Borel equivalence relation on a Polish space X.Let P be the poset for adding a tournament on the E-quotient space as in Exam-ple 8.6.6. Then P is 3, 2-centered by its definition, balanced by Theorem 8.6.4,and tethered by Example 10.5.8. It is compactly balanced, so does not add aE0-transversal by Example 9.2.11. Thus, Theorem 13.2.7 applies to it, as do allother theorems of Section 13.2.


1. Let P be the tournament poset associated with E as in Example 8.6.6. Inthe P -extension of a symmetric Solovay model, there are no discontinuoushomomorphisms of Polish groups, an there is no linear ordering on theE0-quotient space.

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds, thereis a tournament on the E-quotient space while there are no discontinuoushomomorphisms of Polish groups and there is no linear ordering on theE0-quotient space.

Example 13.3.3. Let E be a countable Borel equivalence relation on a Polishspace X. Let P be the poset adding an action of Z on X inducing E as inExample 6.6.10. The poset P is 3, 2-centered by its definition, balanced by Ex-ample 6.6.10, tethered by Example 10.5.7, and it does not add an E0-transversalby Corollary 11.4.12. Thus, Theorem 13.2.7 applies to it, as do all other theo-rems of Section 13.2.

Corollary 13.3.4. Let E be a countable Borel equivalence relation on a Polishspace X.

1. Let P be the poset adding an action of Z on X inducing E as in Exam-ple 6.6.10. In the P -extension of the symmetric Solovay model, there areno discontinuous homomorphisms of Polish groups, and there is no linearordering on the E0-quotient space.

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds, Eis an orbit equivalence relation of a (discontinuous) Z-action, there areno discontinuous homomorphisms of Polish groups, and there is no linearordering on the E0-quotient space.

Example 13.3.5. Let Γ be a Borel graph on a Polish space X Suppose thatΓ does not contain an injective homomorphic copy of K→ω,ω and Kn,ω1

for somenumber n ∈ ω. Let P be the coloring poset of Definition 8.1.1. The poset Pis 3, 2-centered by its definition, it is balanced by Theorem 8.1.2 and tethered


by Example 10.5.4. It does not add a E0-trasnversal by Theorem 11.5.6. Thus,Theorem 13.2.7 applies to it, as do all other theorems of Section 13.2.

Corollary 13.3.6. Let A ⊂ R be a countable set of positive relas converging tozero. Let Γ be the graph on R2 connecting two points if their Euclidean distancebelongs to A.

1. Let P be the coloring poset of Definition 8.1.1. In the P -extension of thesymmetric Solovay model, there are no discontinuous homomorphisms ofPolish groups, and there is no linear ordering of the E0-quotient space.

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds, thechromatic number of Γ is countable, and yet there is no discontinuoushomomorphism between Polish groups and no linear ordering of the E0-quotient space.

The next example shows that the tether assumption is necessary in Theo-rem 13.2.7.

Example 13.3.7. Let P be the collapse poset of |E0| to |2ω| as in Defini-tion 6.6.2. The poset P is 3, 2-centered by its definition and balanced by Corol-lary 6.6.4. It does not add an E0-transversal by Corollary 11.6.3, but it doesadd a linear ordering of the E0-quotient space by mapping it injectively into thelinearly ordered set 2ω.

The following example shows that the assumption that E be pinned is necessaryin Theorem 13.2.7.

Example 13.3.8. Let P be the poset adding a function which selects from eachnonempty countable subset of X = 2ω a single element as in Example 6.6.15.The poset is 3, 2-centered by its definition, balanced by Theorem 6.6.12, tetheredby Example 10.5.6, and adds a linear ordering of the F2-quotient space withoutadding a F2-transversal.

Proof. The poset P does not add an F2-transversal since no balanced poset doesby Corollary 9.1.5. At the same time, an existence of a function f : [X]ℵ0 → Xsuch that for each nonempty countable set a ⊂ X f(a) ∈ a holds implies|F2| ≤ |X<ω1 |. To see the injection from the F2-space toX<ω1 , to each countableset a ⊂ X assign a transfinite sequence 〈xα : α ∈ β by setting recursively xα =f(a\{xγ : γ ∈ α}) unless the set a\{xγ : γ ∈ α} is empty, in which case we finishthe recursion. The set X<ω1 is linearly ordered by the lexicographic ordering,and so the poset P adds a linear ordering of the F2-space.

The following example shows that the pinned assumption in Theorem 13.2.5 isnecessary.

Example 13.3.9. Let E be a pinned Borel equivalence relation on a Polishspace X. Let P be the transversal poset of Example 6.6.8. The poset P is3, 2-centered by its definition and it is balanced by Theorem 6.6.6. Letting Γ

13.3. EXAMPLES 339

be the graph on X connecting any two distinct E-related points, note that Padds a maximal acyclic subgraph to Γ–it is the graph consisting of all edges{x0, x1} ∈ Γ such that one of the vertices x0, x1 belongs to the E-selector addedby P .

Corollary 13.3.10. Let E be a pinned Borel equivalence relation on a Polishspace X.

1. Let P be the transversal poset of Example 6.6.8. In the P -extension of thesymmetric Solovay model, there are no discontinuous homomorphisms ofPolish groups.

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds,E has a transversal, and yet there is no discontinuous homomorphismbetween Polish groups.

The next batch of examples deals with partial orders which are m,n-balancedbut not 3, 2-centered, and violate the conclusions of various preservation theo-rems for 3, 2-centered posets from Section 13.2.

Example 13.3.11. Let Γ be a Borel graph on a Polish space X. Let P bethe poset of all countable acyclic subsets of Γ ordered by reverse inclusion.Whenever n ∈ ω is a natural number, the poset P is n, 2-balanced and everybalanced virtual condition is n, 2-balanced.

The poset P adds a maximal acyclic subset of Γ, violating the conclusion ofTheorem 13.2.5 for many graphs Γ. Therefore, in such cases the poset P is not3, 2-centered and even does not contain a dense analytic 3, 2-centered subset.

Proof. It follows from Example 6.4.10 that balanced virtual conditions for Pare classified by maximal acyclic subgraphs of Γ. Let p be any maximal acyclicsubgraph of Γ. We will check that p is n, 2-balanced, completing the proof.To this end, let {V [Hi] : i ∈ n} be pairwise mutually generic extensions of V ,respectively containing some conditions pi ≤ p for each i ∈ n.

Claim 13.3.12. For each i ∈ n and vertices x0, x1 ∈ X ∩V , the following threeformulas are equivalent:

1. x0, x1 are connected by a path in Γ ∩ V ;

2. x0, x1 are connected by a path in p;

3. x0, x1 are connected by a path in pi.

Proof. (1)→(2) is implied by the maximality of p. (2)→ (3) follows from thefact that p ⊂ pi, and the negation of (1) implies by the Mostowski absolutenessthat x0, x1 are not connected by any path in the graph Γ∩V [Hi], which is largerthan pi and therefore (3) has to fail.


The pairwise mutual genericity implies that if i, j ∈ n are distinct numbers andvi, vj ⊂ X are the sets of vertices mentioned in some edges in pi, pj respectively,then vi∩vj ⊂ V . This, together with the claim, means that the union

⋃i pi can-

not contain a cycle and so is the desired common lower bound of the conditionspi for i ∈ n.

Corollary 13.3.13. Let Γ be a Borel graph on a Polish space X.

1. Write P for the poset adding a maximal acyclic subgraph of Γ as in Ex-ample 6.4.10. In the P -extension of the symmetric Solovay model everyhomomorphism between Polish groups is continuous;

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds, Γhas a maximal acyclic subgraph and every homomorphism between Polishgroups is continuous.

Example 13.3.14. Let F be a Fraisse class of structures in finite relationallanguage with strong amalgamation. Let E be a Borel equivalence relation ona Polish space X. Let P = P (F , E) be the poset of Definition 8.6.3. For eachnumber n ∈ ω, the poset P is n, 2-balanced and every balanced virtual conditionis n, 2-centered.

Note that in case of the linearization poset P for the E0-quotient space, asisolated in Example 8.6.5, is tethered by Example 10.5.8 and does not add anE0-selector by Example 9.2.11. Therefore, P violates the conclusion of Theo-rem 13.2.7, so P contains no dense analytic 3, 2-centered subset.

Proof. It follows from Theorem 8.6.4 that balanced virtual conditions in P areclassified by F-structures on the virtual quotient space X∗∗. Let p be such astructure. Let V [Hi] for i ∈ n be generic extensions of the ground model whichare pairwise mutually generic. Let pi ∈ V [Hi] be a condition extending p, foreach i ∈ n. Note that the domains of the conditions pairwise intersect in thedomain of p. Let b =

⋃i dom(pi). For each finite set a ⊂ b use the strong

amalgamation of the class F n − 1-many times to find a structure Na ∈ F ona so that for each i ∈ n, Na � (dom(pi) ∩ a) = pi � (dom(pi) ∩ a). Let U be anultrafilter on [b]<ℵ0 such that for every finite set c ⊂ b, the set {a ∈ [b]<ℵ0 : c ⊂a} belongs to U . Let N be the structure on b which is the U -integral of thestructures Na. It is immediate that for all N � dom(pi) = pi for each i ∈ 3, andthe closure of the class F under substructures shows that N is a F-structure.It is the desired lower bound of the conditions pi for i ∈ n.


1. Let P be the E-linearization poset as in Example 8.6.5. In the P -genericextension of the symmetric Solovay model every homomorphism betweenPolish groups is continuous;

13.3. EXAMPLES 341

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds,the E-quotient space carries a linear ordering and every homomorphismbetween Polish groups is continuous.

Example 13.3.16. Let X be a Borel vector space over a countable field Φ.Let P be the 3-Hamel decomposition poset of Definition 8.1.11. The poset P is4, 3-balanced.

Theorem 13.2.4 shows that the poset is not 3, 2-balanced as soon as the field Φhas characteristic distinct from 2.

Proof. By Theorem 8.1.12, balanced virtual conditions are classified by color-ings c : X → ω such that any monochromatic triple of pairwise distinct nonzeroelements of X is linearly independent. For each such coloring c, the pairColl(ω,X), c〉 is balanced. We will show that the pair is in fact 4, 3-balanced.

Suppose that in some ambient generic extension, V [Gi] for i ∈ 4 are genericextensions and any three of them are mutually generic. Suppose that pi ∈ V [Gi]for i ∈ 4 are conditions below c; we must show that they have a common lowerbound. Write di = dom(pi) for i ∈ 4. Let d ⊂ X be a countable subspacecontaining

⋃i di as a subset and let a = d \

⋃i di. For each point x ∈ a and

distinct indexes i, j ∈ 4 let eij(x) = {y ∈ di : ∃z ∈ dj : x, y, z are linearlydependent} and as in Claim 8.1.14 argue that each the set eij(x), if nonempty,is a union of Φ-shifts of a single X ∩ V -coset distinct from V .

Now, let J be the Borel ideal on ω used in the definition of P as in Defini-tion 8.1.11. Let b ∈ J be a set which cannot be covered by finitely many sets ofthe form p′′i eij and a finite set. Let b =

⋃{bx : x ∈ a} be a partition of the set b

into sets with the same property. Let q : d→ ω be any map such that⋃i pi ⊂ q

and for each x ∈ a, q(x) ∈ bx \⋃ij p′′i eij(x). Note that the latter union consists

of sets in the ideal J and so the set bx \⋃ij p′′i eij(x) is nonempty. We claim

that q is a common lower bound of the conditions pi for i ∈ 4.

We will only show that any triple {x0, x1, x2} of pairwise distinct nonzerolinearly dependent points is not monochromatic. There are several cases toconsider. If two points of the triple belong to one and the same set di thenso does the third one, then we use the assumption that pi ∈ P is a coloringwithout such monochromatic triples. The case in which each point on the triplebelongs to some di but never to the same one, say x0 ∈ d0 \V , x1 ∈ d1 \V , andx2 ∈ d2 \ V is impossible in view of the assumption that the three extensionsV [G0], V [G1], V [G2] are mutually generic. If one of the points, say x0 belongsto a and the other two do not, say x1 ∈ di and x2 ∈ dj for i, j ∈ 4 distinct,then q(x0) is distinct from all colors in the set p′′i eij(x0) and so is different frompi(x1), and monochromaticity fails again. Finally, if more than one point of thetriple is in the set a, then the triple cannot be monochromatic since the mapq � a is an injection.

Corollary 13.3.17. Let X be a Borel vector space over a countable field Φ.


1. Let P be the 3-Hamel decomposition forcing as in Definition 8.1.11. In theP -generic extension of the symmetric Solovay model every homomorphismbetween Polish groups is continuous;

2. it is consistent relative to an inaccessible cardinal that ZF+DC holds, thereexists a 3-Hamel decomposition of X, and every homomorphism betweenPolish groups is continuous.

Chapter 14

Other combinatorics

The technology of balanced conditions can be applied to prove general theoremsabout lack of other combinatorial objects in the extensions under discourse. Inthis section, we include several theorems that are hopefully elegant axiomatiza-tions and generalizations of earlier results in [63].

14.1 Maximal almost disjoint families

One rather surprising limitation of balanced extensions of the symmetric Solovaymodel is that they contain no maximal almost disjoint families of subsets of ω.The following theorem is stated using Convention 1.7.16.

Theorem 14.1.1. In cofinally balanced extensions of the symmetric Solovaymodel there are no infinite maximal almost disjoint families of subsets of ω.

Proof. Let κ be an inaccessible cardinal. Let P be a Suslin forcing such that Pis balanced below κ. Let W be a symmetric Solovay model derived from κ andwork in W . Suppose towards a contradiction that there is a condition p ∈ P anda name τ such that p forces τ to be an infinite maximal almost disjoint family.The condition p as well as the name τ must be definable from some parameterz ∈ 2ω and some parameters from the ground model. Use Fact 1.7.14 and theassumptions to find an intermediate generic extension V [K] such that z ∈ V [K]and P is balanced in V [K]. Work in the model V [K].

Let p ≤ p be a balanced condition. Let I be the set {a ⊂ ω : for some posetRa of size < κ and some Ra-name σa for a condition in P stronger than p suchthat Ra Coll(ω,< κ) σ P a ∈ τ}.

Claim 14.1.2. ω cannot be covered by a finite set and finitely many elementsof the set I.

Proof. Suppose that J ⊂ I is a finite set such that⋃J ⊂ ω is co-finite. Let

Ha ⊂ Ra for a ∈ J be filters mutually generic over V [K] and let pa = σa/Ha ∈P . By the balance of the condition p, it follows that the set {pa : a ∈ J} ⊂ P

343

344 CHAPTER 14. OTHER COMBINATORICS

has a lower bound, denote it by q. Since the model W is a symmetric Solovayextension of each of the models V [K][Ha], the forcing theorem applied in themodel V [K][Ha] shows that in W , for each a ∈ J , the condition q forces J ⊂ τ .However, since

⋃J ⊂ ω is cofinite, it has to be the case that q J = τ ,

contradicting the initial assumptions on the name τ .

Let U be a nonprincipal ultrafilter on ω disjoint from I. There must be aposet R of size < κ, an R-name η for an infinite subset of ω which is modulofinite included in all sets in U , an R-name χ for a subset of ω and an R-name σ for an element of P stronger than p such that R χ ∩ η is infiniteand R Coll(ω,< κ) σ P χ ∈ τ . This occurs since τ is forced to bea maximal almost disjoint family and therefore must contain an element withinfinite intersection with η.

Now, let H0, H1 ⊂ R be filters mutually generic over the model V [K]. Leta0 = χ/H0, a1 = χ/H1 ∈ P(ω), and p0 = σ/H0, p1 = σ/H1 ∈ P . By thebalance of the condition p ≤ p it must be the case that p0, p1 ∈ P have a lowerbound, denote it by q. Since W is the symmetric Solovay extension of each ofthe models V [K][H0] and V [K][H1], the forcing theorem applied in these modelsshows that W satisfies that q forces both a0 and a1 into τ . The proof will becomplete if we show that a0 6= a1 and a0, a1 have infinite intersection.

For a0 6= a1, observe that neither a0, a1 can belong to the model V [K]. If,say, a0 ∈ V [K] then R witnesses the fact that a0 ∈ I, and consequently a0 musthave both finite and infinite intersection with η/H0, a contradiction. Now, sincea0 ∈ V [K][H0] and a1 ∈ V [K][H1], a mutual genericity argument shows thata0 6= a1.

To establish that the set a0∩a1 is infinite, move back to the model V [K], let〈s0, s1〉 be a condition in the product R × R and n ∈ ω be a number; we mustfind a number m > n and conditions t0 ≤ s0 and t1 ≤ s1 such that t0 m ∈ χand t1 m ∈ χ. To this end, let b0 = {m ∈ ω : ∃t ≤ s0 t m ∈ χ} andb1 = {m ∈ ω : ∃t ≤ s1 t m ∈ χ}. The sets b0, b1 ⊂ ω are both forced to haveinfinite intersection with η and therefore must belong to the ultrafilter U . Thismeans that there is a natural number m > n in the intersection b0 ∩ b1, andthen the desired conditions t0 ≤ s0 and t1 ≤ s1 are found by the definition ofthe sets b0 and b1.

Theorem 8.9.7 above produces a weakly balanced MAD forcing such that in itsextension of the Solovay model, a maximal almost disjoint family exists. As aresult of that example and Theorem 9.1.6, we have

Corollary 14.1.3. It is consistent relative to an inaccessible cardinal thatZF+DC holds, there is an infinite maximal almost disjoint family of subsetsof ω, and there is no uncountable sequence of pairwise distinct reals.

14.2. UNBOUNDED LINEAR SUBORDERS 345

14.2 Unbounded linear suborders

Another general obstruction for reaching ZF independence results in balancedextensions is encapsulated in the following theorem, stated using Convention 1.7.16.

Theorem 14.2.1. Let E be an Fσ-preorder on a Polish space X with no max-imal element such that every countable linearly ordered set has an upper bound.In cofinally balanced σ-closed extensions of the symmetric Solovay model, Econtains no unbounded linearly ordered sets.

A typical preorder satisfying the assumptions is the Turing reducibility preorderor the modulo finite domination ordering on ωω.

Proof. We start with a simple claim which takes place in the context of ZFC. IfR is a poset and η is an R-name for an element of X, say that η is unboundedif R forces that for every ground model element x ∈ X, η E x fails.

Claim 14.2.2. Suppose that R0, R1 are posets and η0, η1 are respective un-bounded names for elements of X. Then R0 ×R1 τ0, τ1 are E-incomparable.

Proof. Let E =⋃n Fn where for every n ∈ ω, Fn is closed. If the conclusion

failed, there would have to be a pair 〈r0, r1〉 ∈ R0 ×R1 forcing say η0 E η1 andthen strengthening the pair if necessary there would have to be a number n ∈ ωsuch that 〈η0, η1〉 ∈ Fn is forced. Let M be a countable elementary submodelof a large structure containing R1, r1, η1, let g ⊂ R1 be a filter generic over themodel M such that r1 ∈ g, and let x = η1/g. Since η0 is an unbounded name,R0 η0 E x fails, and so there must be a condition r′0 ≤ r0 and basic opensets O0, O1 ⊂ X such that (O0 × O1) ∩ Fn = 0, x ∈ O1, and r′0 τ0 ∈ O0.By the forcing theorem, there must be a condition r′1 ∈ g below r1 such thatr′1 τ1 ∈ O1. Then the pair 〈r′0, r′1〉 ≤ 〈r0, r1〉 forces 〈η0, η1〉 ∈ O0 ×O1 and so〈η0, η1〉 /∈ Fn, in contradiction with the initial assumptions.

Now, let κ be an inaccessible cardinal and P be a σ-closed Suslin forcingwhich is balanced cofinally below κ. Let W be the symmetric Solovay modelderived from κ. Suppose towards a contradiction that the conclusion of thetheorem fails in the P -extension of W . Then, in the model W there must bea condition p ∈ P and a P -name τ such that p τ ⊂ X is an E-unboundedlinearly ordered set. The condition p as well as the name τ must be definablefrom some parameter z ∈ 2ω and some parameters in the ground model. LetV [K] be a generic extension of the ground model by a poset of size < κ suchthat p, z ∈ V [K] and P is balanced in V [K].

Work in the model V [K]. Let p ≤ p be a balanced condition in the posetP . Since Coll(ω,< κ) p τ is an unbounded set in ≤, for every elementx ∈ X ∩ V the set τ is forced to contain an element which is not Ex. SinceColl(ω,< κ) p X ∩ V [K] is a countable set and DC holds, τ is forced tocontain a countable subset such that no element of X ∩V [K] is an upper boundof it. By the initial assumptions on the preorder E, this countable set is forcedto have an upper bound in τ , which is then an element of τ which is not Ex


for any element x ∈ X ∩ V [K]. In total, in V [K] there must be a poset R ofsize < κ, an R-name σ for a condition in P such that σ ≤ p and an unboundedR-name η for an element of X such that R η is not E-below any element ofV [K] and R Coll(ω,< κ) σ P η ∈ τ .

In the model W , let H0, H1 ⊂ R be filters mutually generic over V [K].Let p0 = σ/H0 ∈ P and p1 = σ/H1 ∈ P ; by the balance of the conditionp, the conditions p0, p1 are compatible with some lower bound q ∈ P . Letx0 = η/H0 ∈ X and x1 = η/H1 ∈ X; by Claim 14.2.2, these are E-incomparableelements of X. Since the model W is a symmetric extension of both modelsV [K][H0] and V [K][H1], the forcing theorem applied in these models showsthat in W , q x0, x1 ∈ τ . This contradicts the assumption that τ is forced tobe linearly ordered by E.

14.3 Measure and category

In this section, we show that in balanced extensions of the Solovay model, thereis a set of reals without the Baire property. We also provide a frameworkfor showing that in certain circumstances, all sets of reals may be Lebesguemeasurable. The following theorem is stated using Convention 1.7.16.

Theorem 14.3.1. In nontrivial cofinally balanced extensions of the symmetricSolovay model, there is a set of reals without the Baire property.

Proof. Let κ be an inaccessible cardinal. Let P be a nontrivial Suslin poset suchthat Vκ |= P is balanced in every forcing extension. Let W be the symmetricSolovay model derived from κ and work in W . Let p ∈ P be a condition. Wemust find a Polish space X, a condition p ≤ p and a P -name τ for a subset ofX such that p τ does not have the Baire property.

The condition p ∈ P is definable from a real parameter z ∈ 2ω and someparameters in the ground model. Let V [K] be an intermediate extension usinga poset of size < κ such that z ∈ V [K] and P is balanced in V [K], and workin V [K] for a moment. Let p ≤ p be a balanced virtual condition strongerthan P . Choose a poset Q of size < κ and a name σ for a condition in P suchthat the pair 〈Q, σ〉 is balanced and in the balance equivalence class of p. Moveto the Q-extension. Since the poset P is balanced, it is not c.c.c. below σ byProposition 5.2.8(1), in particular it is not Suslin σ-linked below σ. Thus, theanalytic graph of incompatibility of conditions in P below σ has uncountableBorel chromatic number. By the G0-dichotomy, there is a continuous maph : 2ω → P which is a homomorphism of G0 to the incompatibility graph on Pbelow σ. Let S be the Cohen forcing on 2ω introducing a point ygen ; let pgen

be the Q× S-name for h(ygen).Now, back in W , consider the space X of all filters on the two-step iteration

Q× S generic over the model V [K], viewed as a subset of P(Q× S). Since theproduct is countable in W , its powerset gets the usual zero-dimensional compacttopology. Since the model V [K] contains only countably many open densesubsets of the product, the set X ⊂ P(Q×S) is Gδ and therefore Polish. Let τ

14.3. MEASURE AND CATEGORY 347

be the P -name for the set of all filters g ∈ X such that the condition pgen/g ∈ Pbelongs to the generic filter on P ; more formally, τ = {〈g, pgen/g〉 : g ∈ X}. Weclaim that p τ does not have the Baire property.

To show this, first argue that p τ ⊂ X is not meager. Suppose towardsa contradiction that this fails. Then in the model V [K], there exists a posetR of size < κ and R-names σ for a condition in P stronger than p and η fora dense Gδ subset of X such that R Coll(ω,< κ) σ P τ ∩ η = 0. LetG ⊂ Q × S and H ⊂ R be mutually generic filters. By a mutual genericityargument, G ∈ η/H holds. By a balance argument, the conditions σ/H andpgen/G are compatible in P . The common lower bound of these two conditionsforces in P that G ∈ η/H ∩ τ . This is a contradiction.

We will now argue that p τ is not comeager in any nonempty open subsetof X. Suppose towards a contradiction that this fails. Then in the model V [K],there must be a condition 〈q, s〉 ∈ Q × S, a poset R of size < κ and R-namesσ for a condition in P stronger than p and η for a dense Gδ subset of X suchthat R Coll(ω,< κ) σ P ∀g ∈ η 〈q, s〉 ∈ g → g ∈ τ . Let L ⊂ Q andH ⊂ R be mutually generic filters, with q ∈ L. A standard argument provides(in a further generic extension) two G0-related points y0, y1 ∈ 2ω extending swhich are separately generic over the model V [K][H][L]. Let G0, G1 ⊂ Q × Sbe the filters given by L, y0 and L, y1 respectively; both contain the condition〈q, s〉. By the product forcing theorem, the filters G0, H are mutually genericover V [K] and so are the filters G1, H. By a mutual genericity argument, bothfilters G0, G1 both belong to the set η/H. By a balance argument, the conditionpgen/G0 is compatible with σ/H in the poset P . Their common lower boundforces in the poset P that G1 ∈ η/H \ τ . This is a contradiction.

Question 14.3.2. Does the conclusion of Theorem 14.3.1 remain in force inweakly balanced extensions of the symmetric Solovay model?

Other regularity properties of sets of reals may be preserved in the balancedextensions. Consider the following.

Definition 14.3.3. Let Q be a c.c.c. Suslin forcing. Let P be a Suslin forcing.We say that P is Q-balanced if for every condition p ∈ P there is a balancedvirtual condition p ≤ p such that Q p is still balanced. Similar terminologyapplies for Q-weakly balanced forcings.

For the following theorem, let X be a Polish space, let xgen be a layered Q-namefor an element of X, and let I be the σ-ideal of Borel subsets B ⊂ X such thatQ xgen /∈ B. The theorem is stated using Convention 1.7.16.

Theorem 14.3.4. In Q-weakly balanced extensions of the symmetric Solovaymodel, for every set A ⊂ X there is a Borel set B ⊂ X such that A∆B ∈ I.

Proof. Let κ be an inaccessible cardinal. Let P be a Suslin poset such that Pis Q-balanced below κ. Let W be the symmetric Solovay model derived fromκ and work in W . Let p ∈ P be a condition and τ be a P -name such that


p τ ⊂ X. We must find a Borel set B and a stronger condition p ≤ p in Psuch that p τ∆B ∈ I.

The condition p ∈ P and the name τ are definable from a real parameterz ∈ 2ω and some parameters in the ground model. Let V [K] be an intermediateextension using a poset of size < κ such that z ∈ V [K] and work in the modelV [K]. Since P is Q-balanced in V [K], it must be the case that there is aweakly balanced condition p ≤ p such that Q p is weakly balanced. Bya balance argument, Q Coll(ω,< κ) p decides the statement xgen ∈ τ .Let A = A0 ∪ A1 be a maximal antichain of Q such that for every conditionq ∈ A0, q Q Coll(ω,< κ) p P xgen /∈ τ , and for every condition q ∈ A1,q Q Coll(ω,< κ) p P xgen ∈ τ . In W , let B = {x ∈ X : ∃g ⊂ QV [K] : g isgeneric over V [K], contains a condition in A1, and x = xgen/g}. A complexitycalculation in Proposition 2.8.5(1) shows that the set B is Borel since the modelV [K] contains only countably many maximal antichains of Q. It will be enoughto show that p τ∆B ∈ I.

On one hand, it is clear that B ⊂ τ must hold. Whenever x ∈ B is a pointand g ⊂ Q is a filter generic over V [K] containing a condition in A1 and xgen/g =x, then by the forcing theorem in the model V [K], V [K][g] |= Coll(ω,< κ) px ∈ τ , and then by the forcing theorem in V [K][g], W |= p x ∈ τ . By asimilar reasoning, it must be the case that τ \ B ⊂ C where C = {x ∈ X :for no filter g ⊂ QV [K] generic over V [K], xgen/g = x}. However, it is clearthat C ∈ I holds: if there were a condition q ∈ Q forcing xgen ∈ B, taking anyfilter G ⊂ Q generic over W containing the condition q, the point xgen/G mustbelong to C. However, since the poset Q is c.c.c. and Suslin, the filter G∩V [K]is a filter on QV [K] generic over V [K] and so the point x = xgen/G∩V [K] doesnot belong to C, which is a contradiction.

Example 14.3.5. [81] It is consistent relative to an inaccessible cardinal thatZF+DC holds, every set of reals is Lebesgue measurable, and not every set ofreals has the Baire property. For this, use the P -extension of the Solovay modelposet where the Suslin poset P is as in Example 8.8.4 and the Suslin c.c.c. posetQ of closed subsets of R of positive Lebesgue measure ordered by inclusion, withits associated name for the generic real. It is clear that the associated ideal Iis just the ideal of Lebesgue null sets. Now, the balanced conditions for P havebeen classified in Theorem 8.8.2. Let p = 〈ap, bp〉 be a condition. Then the(collapse name for the) pair p = 〈ap, c〉 is a balanced condition whenever c isa dominating set of functions in ωω in the ground model. Now, the poset Qis bounding, therefore the set bp remains dominating in the Q-extension and pis a balanced condition in the Q-extension as well. By Theorem 14.3.4, in theP -extension of the Solovay model, every set of reals is Lebesgue measurable.In that extension, there must be a set of reals without the Baire property byTheorem 14.3.1; in fact, the generic set must fail to have the Baire property.

Example 14.3.6. It is consistent relative to an inaccessible cardinal that ZF+DCholds, every set of reals is Lebesgue measurable, and there is a maximal almostdisjoint family. For this, use the P -extension of the symmetric Solovay model

14.4. DEFINABLY BALANCED FORCING 349

where P is the MAD forcing of Definition 8.9.6. It will be enough to showthat P is Q-weakly balanced, where Q is the Suslin c.c.c. poset Q of closedsubsets of R of positive Lebesgue measure ordered by inclusion. Now, certainweakly balanced conditions for P have been discovered in Theorem 8.9.7. Letp = 〈ap, bp〉 be a condition. Then any (collapse name for a) pair p = 〈ap, c〉is a weakly balanced condition where c is the set of all pairs 〈ap, s〉 where s isa partition of ω into finite intervals in the ground model. Now, let G ⊂ Q bea generic filter, and in the extension V [G], consider any (collapse name for a)pair p = 〈ap, d〉 where d is the set of all pairs 〈ap, s〉 where s is a partition of ωinto finite intervals in V [G]. By Theorem 8.9.7 applied in V [G], p is a weaklybalanced condition in V [G]. Since the poset Q is bounding, for every partitions ∈ V [G] of ω into finite intervals there is a partition t ∈ V of ω such thatevery interval in t contains a subinterval in s. It follows immediately from thedefinition of the poset P that p, p are inseparable in the poset P ; in particular,p is still a weakly balanced condition in V [G].

14.4 Definably balanced forcing

This section is devoted to the class of apparently the softest balanced Suslinextensions one can find: those in which the balanced virtual conditions can befound in an easily definable way.

Definition 14.4.1. Let P be a Suslin forcing. We say that P is definablybalanced if for each condition p ∈ P there is a balanced virtual condition p ≤ pwhich is definable from a real parameter.

Example 14.4.2. Let E be a Borel equivalence relation on a Polish spaceX the poset P of Example 6.6.9 adding a countable complete section of E isdefinably balanced. Namely, let Y = Xω, let B ⊂ Y be the set of those elementsy ∈ Y whose range consists of pairwise E-related points, and let P consist of allcountable sets p ⊂ B such that if y0, y1 ∈ P satisfy [rng(y0)]E = [rng(y1)]E thenin fact rng(y0) = rng(y1) holds. The ordering on P is that of inclusion. Forevery condition p ∈ P one can look at the definable balanced virtual conditionp ≤ p represented by the pair 〈Coll(ω,X), τ〉 where τ is the name for the set ofall conditions q ∈ Q such that p ⊂ q and for every point x ∈ V \

⋃y∈p[rng(y)]E ,

q contains an enumeration of the set [x]E ∩ V .

Example 14.4.3. Let P be the poset of Section 8.4 adding a cofinal Kurepafamily on a fixed Polish space X. For every condition p ∈ P (a countable setof countable subsets of X closed under intersection) one can find a balancedvirtual condition p ≤ p represented by the pair 〈Coll(ω,X), τ〉 where τ is thename for the condition obtained from p by adding the set X ∩V . This is indeeda balanced pair as proved in Theorem 8.4.3, and it is definable from a real.

Example 14.4.4. Let P be the partial ordering introducing a nontrivial au-tomorphism of the algebra P(ω) modulo finite as in Section 8.3. As proved in


Theorem 8.3.3, the balanced virtual conditions are classified precisely by auto-morphisms of the algebra. At the same time, every condition ( an automorphismof a countable subalgebra) can be extended into a total automorphism which istrivial, i.e. generated by a bijection between cofinite subsets of ω [9, Theorem2.3]. Such a total automorphism is clearly definable from a real.

Example 14.4.5. Let Γ be the clopen graph on 2ω connecting points x0 6= x1

if the smallest number n such that x0(n) 6= x1(n) is even. Let P be the partialorder of Example 12.2.16 forcing a failure of OCA by adding an uncountablesubset of 2ω containing no uncountable Γ-clique or Γ-anticlique. For everycondition p ∈ P (which is a pair 〈ap, bp〉 where ap ⊂ 2ω is a countable set andbp is a countable collection of closed Γ-cliques and closed Γ-anticliques) one canfind a balanced virtual condition p ≤ p represented by the pair 〈Coll(ω, 2ω), τ〉where τ is the name for the pair 〈ap, bp〉 where bp is the set of all closed Γ-cliques and all closed Γ-anticliques coded in the ground model. This is indeeda balanced pair below p as shown in Example 12.2.16 and it is definable from areal.

The preservation theorems we prove are best organized with a well-known fac-torization fact and its corollary.

Fact 14.4.6. Let Q be the finite support product of ω1-many copies of the Cohenforcing. Let G ⊂ Q be a generic filter.

1. Whenever x ∈ V [G] is an element of 2ω, V [G] is a Q-extension of V [x].

2. Suppose that X,Y are Polish spaces in V , E is a pinned Borel equivalencerelation on Y coded in V . In V [G], if x ∈ X is a point and c ⊂ Y is anE-class definable from x and some elements of the ground model, then chas a representative in V [x].

The following results are stated using Convention 1.7.16.

Theorem 14.4.7. In definably balanced extensions of the symmetric Solovaymodel, whenever Γ is an analytic hypergraph of countable, possibly infinite ar-ity on a Polish space X of uncountable Borel chromatic number, then Γ hasuncountable chromatic number.

Proof. In view of the dichotomy theorem for uncountable Borel chromatic num-ber of analytic hypergraphs of countable arity [66] it is enough to verify that theconclusion of the theorem holds for a specific analytic hypergraph Γ. Let X bethe dense Gδ subspace of ωω consisting of all functions x ∈ ωω such that for in-finitely many n ∈ ω, x(n) > x(m) for all m ∈ n. Let {sn : n ∈ ω} be a collectionof finite strings of natural numbers such that sn ∈ ωn and {sn : n ∈ ω} ⊂ ω<ω

is dense. Define the hypergraph Γ to consist of all tuples 〈xm : m ∈ ω〉 such thatthere is n ∈ ω such that sn is an initial segment of all points xm, xm(n) = mholds for all m, and the tails xm � (n, ω) are the same for all m ∈ ω. [66] showsthat the graph Γ has uncountable Borel chromatic numbers and moreover, it

14.4. DEFINABLY BALANCED FORCING 351

can be continuously homomorphically mapped to any analytic hypergraph ofuncountable Borel chromatic number. Thus, it is enough to prove the conclu-sion of the theorem for Γ.

Suppose that κ is an inaccessible cardinal. Suppose that P is a Suslin forcingsuch that P is definably balanced below κ. Let W be the symmetric Solovaymodel derived from κ and work in W . Let p ∈ P be a condition and let τ bea P -name for a function from X to ω; we must find a Γ-edge 〈xm : m ∈ ω〉, anumber k ∈ ω and some strengthening of the condition p which forces τ(xm) = kfor all m ∈ ω. The condition p as well as the name τ are defined from someparameters in the ground model and perhaps an additional parameter z ∈ 2ω.Find an intermediate extension V [K] of the ground model using a poset of size< κ such that z ∈ V [K], and work in the model V [K].

Let Q be the product of uncountably many copies of Cohen forcing. By thedefinable balance assumption, there must be a Q-name η for a real number and acondition q ∈ Q which identifies a definition of a balanced virtual condition p ≤ pwhich uses only η as a parameter. By Fact 14.4.6, passing to an intermediateextension if necessary we may assume that η is in fact a name for a specificelement of the ground model. For the sake of brevity, we ignore η and q below.By a balance argument with p, Q ∀x ∈ X ∃kColl(ω,< κ) p P τ(x) = k.Let h be the Q-name for the map x 7→ k; thus, h is forced to be a definableΓ-coloring.

Consider the Cohen poset PX with its name xgen for a generic point of thespace X. Find a condition O ∈ PX and a number k ∈ ω such that O Q h(xgen) = k. Now, let x ∈ O be a point PX -generic over V [K]. Use a genericityargument to find a number m large enough that sm is an initial segment of xand [sm] ∩X ⊂ O. For each number n ∈ ω let xn ∈ X be the point obtainedfrom x by overwriting its m-th entry with n. Thus, each point xn is Q0-genericover V [K] and belongs to the set O. Now, let G ⊂ Q be a filter generic overthe model V [K] such that x ∈ V [K][G], and consider the definable balancedcondition p ≤ p in V [K][G]. By the forcing theorem applied in the model V [K],in the model W p P τ(xn) = k must hold for every number n. In view of thefact that the sequence 〈xn : n ∈ ω〉 is a Γ-hyperedge, the proof is complete.

Corollary 14.4.8. 1. Let P be the poset for adding a nontrivial automor-phism of the algebra P(ω) modulo finite of Section 8.3. In the P -extensionof the Solovay model, there is no discontinuous homomorphism betweenPolish groups.

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds,there is a nontrivial automorphism of P(ω) modulo finite, and there isno discontinuous homomorphism between Polish groups.

Proof. The poset P is definably balanced by Example 14.4.4. It is clearly σ-closed, so in the P -extension of the Solovay model, ZF+DC holds. By Theo-rem 14.4.7, in the P -extension the chromatic number of the Hamming graph H2

is uncountable.. By a ZF+DC result of [76], this abstractly implies that thereare no discontinuous homomorphisms between Polish groups.


Theorem 14.4.9. In definably balanced extensions of the symmetric Solovaymodel, |E1| 6≤ |F | holds for every pinned Borel orbit equivalence relation F .

Proof. Let Γ be a Polish group continuously acting on a Polish spaceX, inducingan equivalence relation F . Suppose that F is pinned. Suppose that P is aSuslin forcing. Suppose further that κ is an inaccessible cardinal such that Pis definably balanced below κ. Let W be the symmetric Solovay model derivedfrom κ and work in W . Suppose that p ∈ P is a condition and τ is a P -namesuch that p τ is a function from the E1-quotient space to the F -quotient space.We must find a strengthening of the condition p and two distinct E1-classes suchthat their τ -images are forced to be the same.

The condition p as well as the name τ are definable from a parameter z ∈ 2ω

as well as some parameters from the ground model. Let V [K] be an intermediateextension obtained from a poset of size < κ such that z ∈ V [K]. Work in themodel V [K]. Let Q be the finite support product of uncountably many copiesof the Cohen forcing. Use the definable balance assumption to find a Q-nameη for a real and a condition q ∈ Q which identifies a definition of a virtualbalanced condition p ≤ p with a parameter η. By Fact 14.4.6, passing into alarger intermediate extension, we may assume that η is in fact a name for aspecific element of V [K]. For the brevity of notation, we neglect η and q below.

Let H ⊂ Q be a filter generic over V [K] and work in the model V [K][H]for the moment. Note that the equivalence relation F is pinned there by The-orem 2.7.1. By a balance argument with p, for every E1-class c in the modelV [K][H] there is an F -class d such that Coll(ω,< κ) p P τ(c) = d. Go backto V [K]. Fix a Q-name h for the map c 7→ d. Note that h is forced to be adefinable injection from the E1-quotient space to the F -quotient space.

Let y ∈ (2ω)ω be a sequence of points in 2ω generic over V [K] for the finitesupport product of the usual Cohen forcing. Working in the model V [K][y], byFact 14.4.6, there is a point x ∈ X such that Q h([y]E1) = [x]F . Let γ ∈ Γ bea point generic over V [K][y] for the poset PΓ. As in the proof of Theorem 4.1.1,V [K][γ ·x] contains no entries of the sequence y, and therefore no representativesof the E1-class of y. Since the class [y]E1

is definable from γ ·x as the class whichis mapped by h to [γ · x]F , this contradicts Fact 14.4.6(2).

Using Example 14.4.2, we now get

Corollary 14.4.10. 1. Let P be the partial ordering adding a countable com-plete section to E1. In the P -extension of the symmetric Solovay model,|E1| 6≤ |F | holds for every pinned orbit equivalence relation F .

2. It is consistent relative to an inaccessible cardinal that ZF+DC holds,|E1| ≤ |F2| and yet |E1| 6≤ |F | for any pinned Borel orbit equivalencerelation F .

14.5 F2 structurability

The following theorem is stated using Convention 1.7.16.

14.5. F2 STRUCTURABILITY 353

Theorem 14.5.1. In σ-closed, balanced, ℵ0-tethered extensions of a symmetricSolovay model there are no tournaments on the F2-quotient space.

The σ-closure demand can be relaxed to include the formally not σ-closed posetsof Section 8.1; we omit the details. The role of ℵ0-tether is less clear. The lin-earization poset on the F2-quotient space of Example 8.6.5 is σ-closed, balanced,and adds a tournament, so clearly some additional assumption beyond the clo-sure and balance is necessary. However, ℵ0-tether is not preserved under takinga regular subposet, while the conclusion of the theorem is. Thus, there is aroom for improvement in the statement of the theorem.

Proof. Let κ be an inaccessible cardinal. Let P be a Suslin poset which is σ-closed, and balanced and ℵ0-tethered below κ. Let W be a symmetric Solovaymodel derived from κ and work in W . Suppose towards a contradiction thatp ∈ P is a condition and τ is a P -name such that p τ is a tournament onthe F2-quotient space. Both p, τ must be definable from some real parameterz ∈ 2ω and some parameters in the ground model. Let V [K] is an intermediateextension by a poset of size < κ such that z ∈ V [K].

Work in the model V [K] for a moment. Let p ≤ p be a balanced virtualcondition in the poset P in V [K]. Let λ < κ be a cardinal such that p isrepresented on a poset of size < λ, let Q0 be the finite support product ofcopies of Coll(ω, λ) indexed by λ+ with finite product of copies of the Cohenforcing indexed by 2× λ+. for any ordinal α ∈ λ+ write Qα0 for the part of theproduct indexed by ordinals below α. Let η0 and η1 be the Q0-names for thesets of Cohen generic reals indexed by {0}×λ+ and {1}×λ+ respectively. In theQ0-extension of V [K] consider the poset Q1 which is the poset P of conditionswhich are stronger than some realization of p. Let Q denote the iteration Q0∗Q1,and let χ be a Q-name for the virtual condition in P consisting of all conditionsin P stronger than all conditions in the generic filter on Q1.

Claim 14.5.2. Q forces χ to be a balanced condition in P below p.

Proof. Let G ⊂ Q0 and H ⊂ Q1 be generic filters over V [K] and work inV [K][G][H]. First, use the σ-closure assumption on P to conclude that in factχ/G ∗ H is forced to be a nonzero virtual condition. In some further collapseextension, the filter H contains a cofinal countable sequence, which then has alower bound and that lower bound will be a condition below χ/G ∗H.

Second, use the tether assumption to conclude that χ/G ∗H is a balancedcondition. Observe that by a genericity argument with the filter H, for everyanalytic subset A ⊂ P coded in the V [K][G0], H contains a condition which iseither below some element of A or incompatible with all elements of A. Sincethe poset P (and so Q1) is σ-closed in V [K][G], it does not add any new analyticsubsets of P . As a result, it is forced that χ/G ∗ H is either stronger than orincompatible with any given virtual condition in P in the model V [K][G][H]carried by a poset of size < ℵ0 (i.e. an analytic subset of P ). The balance ofχ/G ∗H immediately follows from the definition of ℵ0-tether.


Since χ is forced to be balanced, we also have Q Coll(ω,< κ) χ decidesthe statement 〈η0, η1〉 ∈ τ . Let q ∈ Q be a condition such that q Q Coll(ω,<κ) χ P 〈η0, η1〉 ∈ τ (or vice versa). Let q = 〈q0, q1〉 be a breakdown of thecondition q in the iteration Q0 ∗ Q1. Use the λ+-c.c. of the poset Q0 to find anordinal α ∈ λ+ such that q0 is a condition in Qα0 and q1 is a Qα0 -name.

Claim 14.5.3. There are filters G0, G1 ⊂ Q0 which are separately generic overV [K] such that

1. G0 ∩Qα0 and G1 ∩Qα0 are mutually generic over V [K], both containing q;

2. V [K][G0] = V [K][G1]

3. η0/G0 = η1/G1 and η1/G0 = η0/G1.

Proof. Let I = λ∪ (2×λ). Consider any bijection π : I → I such that π′′λ = λ.Such a bijection induces an automorphism of the poset Q and the class of Q-names, which by an abuse of notation we denote by π again. If in additionwe assume that π′′{0} × λ = {1} × λ and π′′{1} × λ = π{0} × λ, then theautomorphism switches the names η0 and η1. If in addition we assume thatπ′′α is disjoint from α and π′′(2×α) is disjoint from 2×α, then the conditionsq, π(q) are compatible. Let G0 ⊂ Q0 be a filter generic over V [K] meeting thelower bound of q and π(q), and let G1 = (π−1)′′G0. The filters G0, G1 are asrequired.

Now, let G0, G1 be filters as in the claim. Let A0 = η0/G0 = η1/G1 andA1 = η1/G0 = η0/G1. The conditions p0 = q/G0 and p1 = q/G1 are compatiblein the poset P by (1), and their lower bound must exist in the model V [K][G0] =V [K][G1]. Let H ⊂ P be a filter generic over V [K][G0] meeting that lowerbound and consider the balanced virtual condition ¯p = χ/G0 ∗ H, which isequal to χ/G1 ∗H. Now, since the filter G0 ∗H ⊂ Q contains the condition q,the forcing theorem shows that in the model W , ¯p 〈A0, A1〉 ∈ τ . By the sameargumentation, since the filter G1 ∗H ⊂ Q contains the condition q, the forcingtheorem shows that in the model W , ¯p 〈A1, A0〉 ∈ τ . This is a contradictionconcluding the proof.

Corollary 14.5.4. 1. Let P = P(ω) modulo finite. In the P -extension ofthe symmetric Solovay model, there is no tournament on the quotient spaceof F2;

2. it is consistent relative to an inaccessible limit of inaccessibles that ZF+DCholds, there is a nonprincipal ultrafilter on ω, and yet there is no tourna-ment on the quotient space of F2.

Theorem 14.5.1 provides many other corollaries, among which we quote thefollowing application of Example 10.5.8:

Corollary 14.5.5. Let E be a pinned Borel equivalence relation on a Polishspace X.

14.6. THE RAMSEY ULTRAFILTER EXTENSION 355

1. Let P be the linearization poset for E as in Example 8.6.5. In the P -extension of the Solovay model, there is no tournament on the F2-quotientspace;

2. it is consistent relative to an inaccessible cardinal that ZF+DC holds, theE-quotient space is linearly ordered, and yet there is no tournament onthe F2-quotient space.

Proof. The linearization poset is certainly σ-closed. It is balanced by Exam-ple 8.6.5 and ℵ0-tethered by Example 10.5.8.

14.6 The Ramsey ultrafilter extension

Let P be the poset of infinite subsets of ω ordered by inclusion. Let κ be aninaccessible cardinal, and let W be the symmetric Solovay model derived from κ.Let U ⊂ P be a filter generic over W . The model W [U ] is a balanced extensionof W by Theorem 7.1.2. It is immediate that U is a nonprincipal ultrafilteron ω and in fact a Ramsey ultrafilter. The model W [U ] has been investigatedfor decades ??? We have provided new information about the model at severallocations in this book. In this section, we outline several major areas of study ofthe model W [U ] and gather the available known results as well as critical openquestions.

Question 14.6.1. Classify the Borel equivalence relations E,F on Polish spacessuch that in W [U ], |E| ≤ |F | holds. In particular, are there Borel equivalencerelations E,F such that |E| ≤ |F | fails in the symmetric Solovay model W andholds in W [U ]?

We have proved that in W [U ], the smooth divide is preserved (Corollary 9.2.5,the orbit divide is preserved (Corollary 9.4.8), and the EKσ -divide is preserved(Corollary 9.5.10. However, our efforts stalled in other directions. We do notknow if the turbulent divide is preserved. We do not know if there are any non-hyperfinite countable Borel equivalence relations E such that |E| ≤ E0| holds inW [U ]; in principle, this inequality could hold for all countable Borel equivalencerelations whatsoever.

Question 14.6.2. Classify the Borel equivalence relations E such that the E-quotient space is linearly ordered in W [U ].

As a basic affirmative result in this direction, we show that E0 and E1-quotientspaces are linearly ordered in W [U ]. To see this, observe in ZF+DC plus theexistence of a nonprincipal ultrafilter on ω that the class of linearly orderableequivalence relations is closed under countable increasing unions. For the proof,let X be a Polish space and 〈En : n ∈ ω〉 be an increasing sequence of analyticequivalence relations on X, each with linearly orderable quotient space. Usethe DC assumption to pick linear orders ≤n for each of them. Let U be anonprincipal ultrafilter on ω. If x, y ∈ X are E-unrelated elements, then let


[x]E ≤ [y]E if the set a(x, y) = {n ∈ ω : [x]En ≤n [y]En} belongs to the ultrafilterU . If x′ E x and y′ E y are different representatives of the E-classes of x, y,then there is a number m ∈ ω such that x Em x′ and y Em y′; therefore, thesymmetric difference a(x, y)∆a(x′, y′) is a subset of m, and the definition of ≤does not depend on the choice of the equivalence class representatives. It is easyto check that ≤ is a linear ordering on the E-classes.

As a basic negative result, the F2-quotient space is not linearly ordered inW [U ]; it even carries no tournament by ???. Another negative result, Theo-rem 14.6.5 below, shows that the E2-quotient space cannot be linearly ordered.As a basic open instance, choose any non-hyperfinite countable Borel equiva-lence relation E.

Question 14.6.3. Let X be an uncountable Polish space. Classify the Borelsets A ⊂ [X]ℵ0 such that in W [U ], there is a function f assigning to each seta ∈ A an ultrafilter on a.

As a basic affirmative result in this direction, if E is a hyperfinite countable Borelequivalence relation on X, then there is an assignment of ultrafilters to E-classesin W [U ]: Let En for n ∈ ω be equivalence relations with all classes finite formingan inclusion-increasing sequence such that E =

⋃nEn. For the first item, let ≤

be any linear ordering ofX. For each point x ∈ X let x(n) be the≤-least elementof X which is En-related to X. Let Ux = {a ⊂ [x]E : {n ∈ ω : x(n) ∈ a} ∈ U}.Thus, Ux is an ultrafilter on [x]E and x E y implies that Ux = Uy since Uis nonprincipal and for some m ∈ ω x Em y holds, and then for all n > mx(n) = y(n) holds.

As a basic negative result in this direction, there is no assignment of ultra-filters to all countable subsets of X; this follows from ???. As a basic openinstance, let Γ be the free group on two generators acting on 2Γ by shift and letX ⊂ Γ be the Gδ set of points on which Γ acts freely. Let E be the orbit equiv-alence relation on X. Is there an assignment of ultrafilters to E-classes? Thisseemingly arbitrary question relates to the quotient cardinal question 14.6.1. Itis well-known that E is not Borel reducible to E0 ???. However. the assignmentof ultrafilters to Γ-orbits yields in ZF the cardinal inequality |E| ≤ |E0|. To

see this, let ∆ be the acyclic locally finite Cayley graph on X and let ~∆ be theorientation of the Cayley graph defined by 〈x, y〉 ∈ ~∆ if the set of {z ∈ X : theunique injective Cayley path from x to z includes y} belongs to the ultrafilter

assigned to [x]E . Clearly, every vertex gets outflow one in the orientation ~∆.Now, apply the argument after Example 9.2.15 to conclude that |E| ≤ |E0| musthold.

Question 14.6.4. Classify ultrafilters on ω in W [U ]. In particular, for whichBorel ideals I on ω is there an ultrafilter F in W [U ] such that I ∩ F = 0?

As the only nontrivial result in this section, we now prove that there is noultrafilter in W [U ] which is disjoint from the summable ideal.

Theorem 14.6.5. Let P be the poset of infinite subsets of ω ordered by inclu-sion. In the P -extension of the symmetric Solovay model, the complement graphassociated with the summable ideal on ω cannot be oriented.


Proof. Write I for the summable ideal on ω; i.e. I consists of sets a ⊂ ω suchthat the sum Σ{ 1

n+1 : n ∈ a} is finite. Towards a contradiction, let κ be aninaccessible cardinal, let W be a symmetric Solovay model derived from κ, letp ∈ P be a condition and τ be a P -name such that p τ is an orientation of thecomplement graph associated with I. Both p, τ must be definable from somereal parameter z ∈ 2ω and some ground model parameters. Let V [K] be anintermediate extension of the ground model by a poset of size < κ containingthe parameter z, and work in the model V [K].

We will produce a poset Q and a Q-name xgen for an element of 2ω suchthat

(I) below every condition q ∈ Q there are conditions q0, q1 ≤ q and an au-tomorphism π : Q � q0 → Q � q1 such that π(xgen) is forced by q1 to beequal to 1− xgen modulo I;

(II) the ultrafilter added by P still generates an ultrafilter in the P × Q-extension.

Once this is done, let G ⊂ P and H ⊂ Q be mutually generic filters over themodel V [K], with p ∈ G, and work in V [K][G][H]. Let x = xgen/H. Let U bethe ultrafilter on ω generated by the filter G; this is possible by the item (II)above. By Theorem 7.1.2, U is a balanced virtual condition in the poset P inthe model V [K][G][H]. The following is proved by a standard argument usingthe balance of U :

Claim 14.6.6. In the model V [K][G][H], either Coll(ω,< κ) U P 〈x, 1 −x〉 ∈ τ , or Coll(ω,< κ) U P 〈1− x, x〉 ∈ τ holds.

Suppose for definiteness that the former alternative prevails. Find a conditionq ∈ H which forces it, and find conditions q0, q1 ≤ q and an automorphismπ : Q � q0 → Q � q1 as in item (I). Find a filter H0 ⊂ Q generic over V [K][G]meeting the condition q0, and let H1 = π′′H0. Let x0, x1 ∈ 2ω be the points as-sociated with the filters H0, H1. Then the models V [K][G][H0] and V [K][G][H1]are equal, and by the forcing theorem applied in V [K][G], it must be true inW that U P 〈x0, 1 − x0〉 ∈ τ , 〈x1, 1 − x1〉 ∈ τ . This is impossible though asx0 = 1− x1 modulo I and τ is forced to be a tournament.

The remainder of the proof consists of the construction of the poset Q sat-isfying (I) and (II). This is a pure ZFC construction in which the concentrationof measure phenomenon is a critical ingredient. Let 〈Jn : n ∈ ω〉 be a veryfast sequence of successive intervals on ω. The following is the concentration ofmeasure type of demand on the intervals in question that we need. Let µn bethe normalized counting measure on 2Jn . Let dn be the metric on 2Jn definedby dn(x, y) = Σ{ 1

m+1 : x(m) 6= y(m)}. We demand that the following holds forevery n ∈ ω:

(III) for every set a ⊂ 2Jn of µn-mass 1/n, the 2−n-neigborhood of a in 2Jn inthe sense of the metric dn has µn-mass greater than 1/2.


The concentration of measure computations in [73, Theorem 4.3.19] show thatsuch a fast sequence of intervals indeed exists. Now, let Tini be the tree of allfinite sequences t such that for each n ∈ dom(t), t(n) ∈ 2Jn . Let Q be thepartial order of all nonempty trees q ⊂ Tini without endnodes such that foreach m ∈ ω there is nm ∈ ω such that for each t ∈ q of length n > nm, theset {x ∈ 2Jn : ta〈x〉 ∈ q} has µn-mass ≥ m/n. The ordering on Q is that ofinclusion. Let xgen be the Q-name for the concatenation of the trunks of thetrees in the generic filter. We must show that items (I) and (II) above hold forQ and xgen .

Item (I) is where the concentration of measure shows up in force. We willin fact show that whenever q0, q1 ∈ Q are arbitrary conditions then there areconditions q′0 ≤ q0, q

′1 ≤ q1 and an isomorphism π : Q � q′0 → Q � q′1 such that

π(1 − xgen) is forced by q′1 to be modulo I equivalent to xgen . To see this, lett0 ∈ q0, t1 ∈ q1 be nodes of the same length such that for every node t ∈ q0 oflength n ≥ |t0|, the set a0(t) = {x ∈ 2Jn : ta〈x〉 ∈ q0} has µn-mass at least 2/n,and for every node t ∈ q1 of length n ≥ |t0|,the set a1(t) = {x ∈ 2Jn : ta〈x〉 ∈ q1}has µn-mass at least 2/n as well. We will produce a tree q′0 ≤ q0 � t0 in Q anda level and order preserving injection π : q′0 → q1 � t1 so that

(IV) for each node t ∈ q′0 and a number n > |t0| in dom(t), dn(t(n), 1 −π(t)(n)) < 2−n+1.

Write q′1 = π′′q′0. Since the measures µn are normalized counting measures, themap π naturally extends to an isomorphism π : Q � q′0 → Q � q′1. The demand(IV) then implies that π(1− xgen) is forced by q′1 to be modulo I equivalent toxgen as required.

The map π is obtained by a hungry algorithm. Among all level and orderpreserving injections from subsets of q0 � t0 to q1 � t1 which satisfy (IV), se-lect an inclusion maximal one and call it π. It will be enough to show thatq′0 = dom(π) belongs to Q. To see this, first of all q′0 is closed under initialsegment by an obvious maximality argument. To verify the branching condi-tion in the definition of Q for q′0, let t ∈ q′0 of length some n ≥ |t0|. It willbe enough to argue that the set {x ∈ 2Jn : ta〈x〉 ∈ q′0} has µn-mass at leastmin(µn(a0(t)), µn(a1(π(t)))− 1/n.

To do this, note that if both sets b0 = {x ∈ 2Jn : ta〈x〉 ∈ q′0 \ q0} andb1 = {x ∈ 2Jn : ta〈x〉 ∈ π(q′0) \ q1} had µn-mass greater than 1/n, then by(III) the 2−n-neighborhood of b0 and the set {1 − x : x ∈ b1} would both haveµn-mass greater than 1/2 and so they would intersect, making it possible toextend π while satisfying (IV) and contradicting the maximality of π.

Item (II) follows from a density argument on P and Q given the fact that theposet Q is proper and does not add independent reals [101, Theorem 4.4.8].

Corollary 14.6.7.

1. Let P = P(ω) modulo finite. In the P -extension of the symmetric Solovaymodel, there is no ultrafilter on ω disjoint from the summable ideal;


2. it is consistent relative to an inaccessible cardinal that ZF+DC holds, thereis a nonprincipal ultrafilter on ω, and yet there is no ultrafilter disjointfrom the summable ideal.


Bibliography

[1] Francis Adams and Jindrich Zapletal. Cardinal invariants of closed graphs.Israel Journal of Mathematics, 227:861–888, 2018.

[2] Martin Aigner. Combinatorial theory. Grundlehren der mathematischenWissenschaften 234. Springer Verlag, New York, 1979.

[3] Reinhold Baer. Abelian groups that are direct summands of every con-taining abelian group. Bulletin of the American Mathematical Society,46:800–807, 1940.

[4] Bohuslav Balcar, Thomas Jech, and Jindrich Zapletal. Semi-Cohenboolean algebras. Annals of Pure and Applied Logic, 87:187–208, 1997.

[5] John Baldwin and Paul B. Larson. Iterated elementary embeddings andthe model theory of infinitary logic. Annals of Pure and Applied Logic,167:309–334, 2016.

[6] John T. Baldwin, Sy D. Friedman, Martin Koerwien, and Michael C.Laskowski. Three red herrings around Vaught’s conjecture. Trans. Amer.Math. Soc., 368:3673–3694, 2016.

[7] Tomek Bartoszynski and Haim Judah. Set Theory. On the structure ofthe real line. A K Peters, Wellesley, MA, 1995.

[8] James Baumgartner and Adam Taylor. Partition theorems and ultrafil-ters. Transactions of the American Mathematical Society, 241:283–309,1978.

[9] A. Bella, Alan Dow, Klaas-Pieter Hart, Michael Hrusak, Jan van Mill,and P. Ursino. Embeddings of P(ω)/fin and extension of automorphisms.Fundamenta Mathematicae, 174:271–284, 2002.

[10] Andreas Blass. Ultrafilters related to Hindmans finite-unions theoremand its extensions. In Stephen Simpson, editor, Logic and Combinatorics,Contemporary Mathematics 65, pages 89–124. American MathematicalSociety, Providence, 1987.

[11] Andres Blass, Natasha Dobrinen, and Dilip Raghavan. The next bestthing to a P-point. Journal of Symbolic Logic, 80:866–900, 2015.

361

362 BIBLIOGRAPHY

[12] Jorg Brendle, Fabiana Castiblanco, Ralf Schindler, Liuzhen Wu, and LiangYu. A model with everything except a well-ordering of the reals. 2018.arXiv:1809.10420.

[13] Jorg Brendle and Michael Hrusak. Countable Frechet groups: An inde-pendence result. Journal of Symbolic Logic, 74:1061–1068, 2009.

[14] B. Bukh. Measurable sets with excluded distances. Geometric and Func-tional Analysis, 18:668–697, 2008.

[15] Lev Bukovsy. Iterated ultrapower and Prikry’s forcing. Comment. Math.Univ. Carolinae, 18:77–85, 1977.

[16] Andres Caicedo, John Clemens, Clinton Conley, and Benjamin Miller.Definability of small puncture sets. Fundamenta Mathematicae, 215:39–51, 2011.

[17] Jack Ceder. Finite subsets and countable decompositions of Euclideanspaces. Rev. Roumaine Math. Pures Appl., 14:1247–1251, 1969.

[18] David Chodounsky and Osvaldo Guzman. There are no P-points in Silverextensions. Institute of Mathematics, Czech Academy of Sciences preprint17-2017, 2017.

[19] Clinton Conley and Benjamin Miller. A bound on measurable chromaticnumbers of locally finite Borel graphs. Mathematical Research Letters,23:1633–1644, 2016.

[20] Patrick Dehornoy. Iterated ultrapowers and Prikry forcing. Ann. Math.Logic, 15:109–160, 1978.

[21] Carlos DiPrisco and Stevo Todorcevic. Perfect set properties in L(R)[U ].Advances in Mathematics, 139:240–259, 1998.

[22] Natasha Dobrinen and Daniel Hathaway. Forcing and the Halpern–Lauchlitheorem. Journal of Symbolic Logic, 2019. to appear.

[23] Natasha Dobrinen, Jose G. Mijares, and Timothy Trujillo. TopologicalRamsey spaces from Fraisse classes, Ramsey-classification theorems, andinitial structures in the Tukey types of p-points. Archive for MathematicalLogic, 56:733–782, 2017.

[24] Alan Dow. Set theory in topology. In Recent progress in general topology.North Holland, New York, 1992.

[25] Paul Erdos and Andras Hajnal. On chromatic number of graphs and setsystems. Acta Math. Acad. Sci. Hung., 17:61–99, 1966.

[26] Ilijas Farah. Semiselective coideals. Mathematika, 45(1):79–103, 1998.

BIBLIOGRAPHY 363

[27] Ilijas Farah. Ideals induced by Tsirelson submeasures. Fundamenta Math-ematicae, 159:243–258, 1999.

[28] Ilijas Farah. Analytic quotients. Memoirs of AMS 702. American Mathe-matical Society, Providence, 2000.

[29] Matthew Foreman and Menachem Magidor. Large cardinals and definablecounterexamples to the continuum hypothesis. Ann. Pure Appl. Logic,76:4797, 1995.

[30] R. Fraiss e. Sur lextension aux relations de quelques propriet es des ordres.Ann. Sci. Ecole Norm. Sup., 7:363–388, 1954.

[31] Harvey Friedman and Lee Stanley. A Borel reducibility theory for classesof countable structures. J. Symbolic Logic, 54:894–914, 1989.

[32] Su Gao. Invariant Descriptive Set Theory. CRC Press, Boca Raton, 2009.

[33] E. Glasner, B. Tsirelson, and B. Weiss. The automorphism group of theGaussian measure cannot act pointwise. Israel Journal of Mathematics,148:305–329, 2005.

[34] G. Grunwald. Egy halmazelmeleti tetelrol. Mathematikai es Fizikai Lapok,44:51–53, 1937.

[35] Andras Hajnal and A. Mate. Set mappings, partitions, and chromaticnumbers. In Logic Colloquium 1973, pages 347–379. North Holland, Am-sterdam, 1975.

[36] Rudolf Halin. Uber unendliche Wege in Graphen. Mathematische An-nalen, 157:125–137, 1964.

[37] E. Hall, K. Keremedis, and E. Tachtsis. The existence of free ultrafilterson ω does not imply the extension of filters on ω to ultrafilters. Math.Logic Quarterly, 59, 2013.

[38] Philip Hall. On representatives of subsets. J. London Math. Soc., 10:26–30, 1935.

[39] James Henle, Adrian R. D. Mathias, and W. Hugh Woodin. A barren ex-tension. In Methods in mathematical logic, Lecture Notes in Mathematics1130, pages 195–207. Springer Verlag, New York, 1985.

[40] Neil Hindman and Donna Strauss. Algebra in Stone–Cech compactifica-tion: theory and applications. De Gruyter, New York, 1998. De GruyterExpositions in Mathematics 27.

[41] Wilfrid Hodges. A shorter model theory. Cambridge University Press,Cambridge, 1997.

364 BIBLIOGRAPHY

[42] Haim Horowitz and Saharon Shelah. Transcendence bases, well-orderingsof the reals and the axiom of choice. ArXiv:1901.01508, 2019.

[43] M. Hrusak, D. Meza-Alcantara, E. Thummel, and C. Uzcategui. Ramseytype properties of ideals. Annals of Pure and Applied Logic, 168:2022–2049, 2017.

[44] Jaime Ihoda (Haim Judah) and Saharon Shelah. Souslin forcing. TheJournal of Symbolic Logic, 53:1188–1207, 1988.

[45] Thomas Jech. Set Theory. Springer Verlag, New York, 2002.

[46] Haim Judah, Andrzej Roslanowski, and Saharon Shelah. Exam-ples for Souslin Forcing. Fundamenta Mathematicae, 144:23–42, 1994.arxiv:math.LO/9310224.

[47] Anastasis Kamburelis. Iterations of boolean algebras with measure. Arch.Math. Logic, 29:21–28, 1989.

[48] Vladimir Kanovei. Borel Equivalence Relations. University Lecture Series44. American Mathematical Society, Providence, RI, 2008.

[49] Vladimir Kanovei, Marcin Sabok, and Jindrich Zapletal. Canonical Ram-sey Theory on Polish Spaces. Cambridge Tracts in Mathematics 202.Cambridge University Press, 2013.

[50] Itay Kaplan and Saharon Shelah. Forcing a countable structure to belongto the ground model. Math. Log. Q., 62:530–546, 2016.

[51] M. Kaufmann. The quantifier “there exist uncountably many” and someof its relatives. In S. Barwise and J Feferman, editors, Model theoreticlogics, pages 123–176. Springer-Verlag, New York, 1985.

[52] Alexander Kechris. Actions of Polish groups and classification problems,pages 115–187. London Mathematical Society Lecture Note Series 262.Cambridge University Press, Cambridge, 2003.

[53] Alexander Kechris, Vladimir Pestov, and Stevo Todorcevic. Frasse lim-its, Ramsey theory, and topological dynamics of automorphism groups.Geometric and Functional Analysis GAFA, 15:106189, 2005.

[54] Alexander Kechris, Slawomir Solecki, and Stevo Todorcevic. Borel chro-matic numbers. Advances in Mathematics, 141:1–44, 1999.

[55] Alexander S. Kechris. Classical Descriptive Set Theory. Springer Verlag,New York, 1994.

[56] Richard Ketchersid, Paul Larson, and Jindrich Zapletal. Ramsey ultrafil-ters and countable-to-one uniformization, 2016.

BIBLIOGRAPHY 365

[57] Julia Knight, Antonio Montalban, and Noah Schweber. Computable struc-tures in generic extensions. Journal of Symbolic Logic, 81:814–832, 2016.

[58] Peter Komjath. The list-chromatic number of infinite graphs defined onEuclidean spaces. Discrete Comput. Geom., 45:497–502, 2011.

[59] Peter Komjath and James Schmerl. Graphs on Euclidean spaces definedusing trascendental distances. Mathematika, 58:1–9, 2019.

[60] Peter Komjath and Saharon Shelah. Coloring finite subsets of uncountablesets. Proceedings of the American Mathematical Society, 124:3501–3505,1996. arxiv:math.LO/9505216.

[61] Georges Kurepa. A propos d’une generalisation d’ue la notion d’ensemblesbien ordonnes. Acta Math., 75:139–150, 1943.

[62] Adam Kwela and Marcin Sabok. Topological representations. J. Math.Anal. Appl., 422:14341446, 2015.

[63] Paul Larson and Jindrich Zapletal. Canonical models for fragments of theaxiom of choice. Journal of Symbolic Logic, 82:489–509, 2017.

[64] Paul B. Larson. The stationary tower. University Lecture Series 32. Amer-ican Mathematical Society, Providence, RI, 2004. Notes from Woodin’slectures.

[65] Paul B. Larson. Scott processes. In Beyond first order model theory.Volume I. CRC Press, New York, 2017.

[66] D. Lecomte and B. D. Miller. Basis theorems for non-potentially closedsets and graphs of uncountable Borel chromatic number. J. Math. Log.,8:121–162, 2008.

[67] David Marker. Model theory: An introduction. Graduate Texts in Math-ematics 217. Springer Verlag, 2002.

[68] Andrew Marks and Spencer Unger. Borel measurable paradoxical decom-positions via matchings. Advances in Mathematics, 289:397–410, 2016.

[69] A. R. D. Mathias. On sequences generic in the sense of Prikry. J. Austral.Math. Society, 15:409–414, 1973.

[70] Diego Mejıa. Matrix iterations with vertical support restrictions. In Pro-ceedings of the 15thAsian Logic Conference. World Sci. Publ., Hackensack,NJ. to appear.

[71] Jaroslav Nesetril. Ramsey classes and homogeneous structures. Combi-natorics, Probability and Computing, 14:171–189, 2005.

[72] Jaroslav Nesetril and Vojtech Rodl. Partitions of finite relational sets andsystems. Journal of Combinatorial Theory Ser. A, 22:289–312, 1977.

366 BIBLIOGRAPHY

[73] Vladimir Pestov. Dynamics of Infinite-Dimensional Groups. UniversityLecture Series 40. Amer. Math. Society, Providence, 2006.

[74] Christian Rosendal. Cofinal families of Borel equivalence relations andquasiorders. J. Symbolic Logic, 70:1325–1340, 2005.

[75] Christian Rosendal. Automatic continuity of group homomorphisms. Bull.Symbolic Logic, 15:184–214, 2009.

[76] Christian Rosendal. Continuity of universally measurable homomor-phisms. Forum of Mathematics, Pi, 2019. to appear.

[77] Christian Rosendal and S lawomir Solecki. Automatic continuity of ho-momorphisms and fixed points on metric compacta. Israel Journal ofMathematics, 162:349–371, 2007.

[78] Edward R. Scheinerman and Daniel H. Ullman. Fractional graph theory.Wiley Series in Discrete Mathematics and Optimization. John Wiley andsons, New York, 1997.

[79] Ralf Schindler, Liuzhen Wu, and Liang Yu. Hamel bases and the principleof dependent choice. 2018. preprint.

[80] James H. Schmerl. Countable partitions of Euclidean space. Math. Proc.Camb. Phil. Soc., 120:7–12, 1996.

[81] Saharon Shelah. On measure and category. Israel Journal of Mathematics,52:110–114, 1985.

[82] Saharon Shelah. Proper and Improper Forcing. Springer Verlag, NewYork, second edition, 1998.

[83] Saharon Shelah and Juris Steprans. PFA implies all automorphisms aretrivial. Proceedings of the American Mathematical Society, 104:1220–1225,1988.

[84] Saharon Shelah and Jindrich Zapletal. Ramsey theorems for product offinite sets with submeasures. Combinatorica, 31:225–244, 2011.

[85] Slawomir Solecki. Analytic ideals and their applications. Annals of Pureand Applied Logic, 99:51–72, 1999.

[86] Juris Steprans. Strong Q-sequences and variations o Martin’s axiom.Canadian Journal of Mathematics, 37:730–746, 1985.

[87] Jacques Stern. On Lusin’s restricted continuum problem. Annals of Math-ematics, 120:7–37, 1984.

[88] Endre Szemeredi. On sets of integers containing no k elements in arith-metic progression. Acta Arithmetica, 27:199245, 1975.

BIBLIOGRAPHY 367

[89] A. Szymanski and Zhou Hao Xua. The behaviour of ω2∗ under someconsequences of Martin’s axiom. In General topology and its relations tomodern analysis and algebra, V, Prague, 1981, number 3 in Sigma Ser.Pure Math., pages 577–584. Heldermann, Berlin, 1983.

[90] Simon Thomas and Jindrich Zapletal. On the Steinhaus and Bergmanproperties for infinite products of finite groups. Confluentes Math., 4,2012. 1250002.

[91] Stevo Todorcevic. Partition problems in topology. Contemporary Mathe-matics 84. American Mathematical Society, Providence, RI, 1989.

[92] Stevo Todorcevic. Remarks on Martin’s Axiom and the Continuum Hy-pothesis. Canadian Journal of Mathematics, 43:832–851, 1991.

[93] Stevo Todorcevic. Two examples of Borel partially ordered sets with thecountable chain condition. Proceedings of the American MathematicalSociety, 112:1125–1128, 1991.

[94] Stevo Todorcevic. Walks on ordinals and their characteristics. BirkhauserVerlag, Basel, 2007. Progress in Mathematics 263.

[95] Todor Tsankov. Automatic continuity for the unitary group. Proceedingsof American Mathematical Society, 141:3673–3680, 2013.

[96] Douglas Ulrich, Richard Rast, and Michael C. Laskowski. Borel complex-ity and potential canonical Scott sentences. Fund. Math., 239:101–147,2017.

[97] Mark Brian VanLiere. Splitting the reals into two small pieces. PhD thesis,University of California, Berkeley, 1982.

[98] Boban Velickovic. Definable automorphisms of P(ω)/fin. Proceedings ofthe American Mathematical Society, 96:130–135, 1986.

[99] Hugh Woodin. Supercompact cardinals, sets of reals and weakly homo-geneous trees. Proceedings of the National Academy of Sciences USA,85:6587–6591, 1988.

[100] Hugh Woodin. The Axiom of Determinacy, Forcing Axioms and the Non-stationary Ideal. Walter de Gruyter, New York, 1999.

[101] Jindrich Zapletal. Forcing Idealized. Cambridge Tracts in Mathematics174. Cambridge University Press, Cambridge, 2008.

[102] Jindrich Zapletal. Interpreter for topologists. Journal of Logic and Anal-ysis, 7:1–61, 2015.

[103] Jindrich Zapletal. Hypergraphs and proper forcing. Journal of Mathemat-ical Logic, 2019. to appear.

368 BIBLIOGRAPHY

[104] Jindrich Zapletal. Subadditive families of hypergraphs. 2019. unpublished.

[105] Joseph Zielinski. The complexity of the homeomorphism relation betweencompact metric spaces. Advances in Mathematics, 291:635–645, 2016.

[106] Andy Zucker. Big Ramsey degrees and topological dynamics. Groups,Geometry, and Dynamics, 13:235–276, 2019.

Index

absolutenessMostowski, 30Shoenfield, 30

below κ, 33

cardinalλ(E), 50, 152Erdos, 52measurable, 76, 77pinned, κ(E), 3, 50

coloring, 29coloring number

Borel, ℵ1, 137, 214, 222countable, 169, 247, 281

complete countable section, 153complex, 133

algebraic, 246locally countable, 135modular, 214, 246

concentration of measure, 5, 97, 358condition

m,n-balanced, 327balanced, 122placid, 212virtual, 120weakly balanced, 128

decomposition3-Hamel, 172, 226, 341acyclic, 204, 227Hamel, 176

end of graph, 209equivalence

orbit, 15, 112, 217pinned, 42placid, 88

virtually placid, 88equivalence, specific

E0, 6, 28E1, 28, 217E2, 28EΓ, 28, 46Eω1

, 3, 28, 47, 76, 77F2, 4, 28, 77

forcingm,n-balanced, 16, 327m,n-centered, 327balanced, 124Bernstein balanced, 305compactly balanced, 15, 205, 219,

292definably balanced, 349nested balanced, 15, 217perfect, 299perfectly balanced, 16placid, 15, 212, 310pod balanced, 230reasonable, 65Suslin, 119Suslin σ-centered, 266Suslin σ-liminf-centered, 269Suslin σ-linked, 265Suslin σ-Ramsey-centered, 266tethered, 15, 239, 353very Suslin, 261weakly balanced, 131

forcing, specific3-Hamel, 3413-Hamel decomposition, 173, 226E-linearization, 13, 188, 208, 220,

313, 322, 340E,F -collapse, 151, 214, 250, 322

369

370 INDEX

E,F -transversal, 14, 152, 214, 221,248

E,F-Fraisse, 186, 215, 248, 312PX , 4, 32, 81, 84Γ-coloring, 169, 225, 247, 311Γ,∆-homomorphism, 177, 210, 221,

247Γ, n-coloring, 210Coll(ω,< κ), 32Coll(ω,X), 32Qκ, 33, 321P(ω) mod fin, 12, 207, 220, 249,

303, 322, 356acyclic, 145, 225, 322, 339acyclic decomposition, 204, 216automorphism, 179finite-countable, 190Hamel basis, 12, 144, 311, 322Hamel decomposition, 177Kurepa, 250Lusin, 193, 313, 322Lusin collapse, 195, 204MAD, 199, 344matroid, 148, 246, 311, 312

generic ultrapower, 33, 141, 321graph

ΓK, 134G0, 254Hamming on ωω, 286, 287Hamming, diagonal, 287Euclidean distance, 287Hamming, 254, 328, 351Hamming, diagonal, 8, 254, 286,

290Hamming, on ωω, 8, 254

hypergraphactionable, 292equilateral triangle, 10

idealω-hitting, 85branch, 92countably separated, 92, 124P-ideal, 309

Rado graph, 103summable, 10, 101, 356Tsirelson, 103

independent maps, 4, 82, 84, 86, 318

jumpFriedman–Stanley, 42

Kurepa family, 181, 211, 215, 221

large fragment of ZFC, 28large structure, 27

matroid, 143algebraic, 146, 150gammoid, 147graphic, 145linear, 144transversal, 145

modularcomplex, 20, 139function, 139pre-geometry, 143

numberBorel σ-bounded chromatic, 255,

292, 294Borel σ-bounded clique, 260Borel σ-bounded density, 288Borel σ-bounded fractional chro-

matic, 258, 296Borel σ-finite clique, 260countable Borel chromatic, 7, 254,

291fractional chromatic, 256

OCA, 16, 307OCA+, 301

perfect matching, 7, 145, 209, 278pin

E-pin, 37, 88P -pin, 120

pod, 229pre-geometry, 143product

measured skew, 259

INDEX 371

skew, 254

quotient space, 29

sequencechoice-coherent, 109, 216coherent, 5, 106

setcentered, 266liminf-centered, 269linked, 265maximal, 148Ramsey centered, 266strongly maximal, 140

set mapping, 333

tournament, 9, 188, 323transversal, 28turbulence, 4, 84, 212

ultrafilterRamsey, 303stable ordered union, 304

uniformizationpinned, 240Saint-Raymond, 244well-orderable, 242

virtualequivalence class, 38structure, 40

walk, 81, 84, 97

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