Infinite and Finitary Combinatorics AroundHrushovski Constructions

Thesis submitted in partial fulfillment of the requirements for the degree ofDOCTOR OF PHILOSOPHY

Ben-Gurion University of the Negev

Omer Mermelstein

August 12, 2018

Contents

1 Introduction 51.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4.1 Notation and conventions . . . . . . . . . . . . . . . . 101.4.2 Pregeometries . . . . . . . . . . . . . . . . . . . . . . . 12

2 Fraısse-Hrushovski amalgamation classes 172.1 Predimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Self-sufficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3 Amalgamation . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4 The generic structure . . . . . . . . . . . . . . . . . . . . . . . 232.5 Isomorphism of pregeometries . . . . . . . . . . . . . . . . . . 252.6 Reducts of generic structures . . . . . . . . . . . . . . . . . . . 27

3 Hrushovski’s ab initio constructions of varying degrees ofsymmetry 323.1 The structures Mg and their pregeometries . . . . . . . . . . . 32

3.1.1 The class Cg . . . . . . . . . . . . . . . . . . . . . . . . 323.1.2 Isomorphism of the pregeometries of M 6∼ and Mg . . . 35

3.2 Reduction relations between the structures Mg . . . . . . . . . 393.2.1 Reduction of Mh to Mg for h ≤ g . . . . . . . . . . . . 393.2.2 Exquisite formulas . . . . . . . . . . . . . . . . . . . . 413.2.3 Constructing an exquisite formula . . . . . . . . . . . . 483.2.4 Proper reduction of M∼ to M 6∼ . . . . . . . . . . . . . 51

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4 The clique construction and the geometric construction 554.1 The clique construction . . . . . . . . . . . . . . . . . . . . . . 554.2 Mclq is a proper reduct of M 6∼ . . . . . . . . . . . . . . . . . . 574.3 The pregeometry of Mclq . . . . . . . . . . . . . . . . . . . . . 604.4 A two-sorted variation of the construction . . . . . . . . . . . 624.5 The geometric construction . . . . . . . . . . . . . . . . . . . 644.6 Mgeo is the pregeometry of a generic structure . . . . . . . . . 66

5 Closed ordinal Ramsey numbers 725.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.1.1 Description of proof . . . . . . . . . . . . . . . . . . . . 735.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.2.1 Skeletons . . . . . . . . . . . . . . . . . . . . . . . . . 755.2.2 The sets F (ωk)rn . . . . . . . . . . . . . . . . . . . . . . 78

5.3 Reducing a finite pair-colouring to a canonical colouring . . . 795.4 Colouring in two colours . . . . . . . . . . . . . . . . . . . . . 83

5.4.1 Canonical triangle-free graphs . . . . . . . . . . . . . . 845.5 Rcl(ω · 2, 3) = ω3 · 2 . . . . . . . . . . . . . . . . . . . . . . . . 85

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Abstract

The thesis is composed of two distinct and independent parts.

Fraısse-Hrushovski constructions

In the first part of the thesis we investigate relationships between severalvariations and generalizations of Hrushovski’s non-collapsed ab initio con-struction.

In Chapter 2 we go through the definition of a Fraısse-Hrushovski amal-gamation class, its generic structure, and its associated pregeometry. Wedevelop the tool kit that is later used for showing that pregeometries asso-ciated to different Fraısse-Hrushovski amalgamation classes are isomorphic,and for showing that the generic structure of one simple Fraısse-Hrushovskiamalgamation class is isomorphic to a reduct of the generic structure of an-other simple Fraısse-Hrushovski amalgamation class.

In Chapter 3 we present the Fraısse-Hrushovski amalgamation classes Cg

corresponding to Hrushovski’s classic non-collapsed ab initio construction –the class of finite directed n-uniform hypergraphs with hereditarily no moreedges than vertices – and standard generalizations of it, allowing symmetryon the hypergraph’s edges. The predimension function δ accompanying eachsuch class is the one which assigns to each hypergraph its number of verticesminus its number of edges. We show that all these classes have the sameassociated pregeometry and that each of their generic structures is isomorphicto a proper reduct of the generic structure of any of the classes.

In Chapter 4 we introduce λ, a generalization of the predimension func-tion δ of Chapter 3. The function λ, instead of counting edges, counts sizesof cliques. This gives rise to an analogous simple Fraısse-Hrushovski amal-gamation class Cclq, which we prove has the same associated geometry asthat of the classes Cg, and whose generic structure is a reduct of the generic

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structure associated to Cg.We conclude the first part of the thesis with the introduction of Cgeo,

a subclass of Cclq whose elements capture the pregeometries associated toelements of Cg that prohibit large subsets X with predimension lower thann− 1. We show that this is a simple Fraısse-Hrushovski amalgamation classwith respect to λ and that the pregeometry associated to it is in fact itsgeneric construction.

Closed ordinal Ramsey numbers

For ordinals α, β, write Rcl(α, β) for the least ordinal δ such that for everypair-colouring c : [δ]2 → {0, 1}, there exists either a c-homogeneous set X ∈[δ]α, closed in its supremum, of colour 0, or a c-homogeneous set Y ∈ [δ]β,closed in its supremum, of colour 1. In the second part of the thesis we showthat Rcl(ω · 2, 3) = ω3 · 2.

We show that each pair-colouring c : [δ]2 → k, for δ < ωω and k ∈ N,can be reduced to a canonical colouring. The bound Rcl(ω · 2, 3) ≤ ω3 · 2 isachieved by a combinatorial analysis of canonical colourings in two colours.The bound Rcl(ω · 2, 3) ≥ ω3 · 2 is given by an explicit colouring c : [ω3 ·2]2 → {0, 1} with no 1-homogeneous sets of size 3, and such that every0-homogeneous set of order type ω · 2 closed in its supremum is cofinal inω3 · 2.

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Chapter 1

Introduction

The thesis is based on several papers either submitted or to be submittedfor publication, and is divided into two distinct and independent topics. Thefirst part of the thesis spans Chapters 1 to 4 and contains the material of[Mer16], [HM17], [Mer17a], all pertaining to Hrushovski constructions. Thesecond part of the thesis is Chapter 5, a self-contained elementary calculationof a closed ordinal Ramsey number – as appearing in [Mer17b]. We defer theintroduction of the second part to the beginning of Chapter 5.

1.1 Background

The class of combinatorial pregeometries associated to a structure (or a the-ory) is an important invariant in geometric stability theory and its bifur-cations, going back to Baldwin-Lachlan [BL71], Zilber’s Trichotomy con-jecture [Zil84] and its many applications, e.g. [HZ96], Shelah’s analysis ofsuper-stable theories [She90, Chapters V, IX, X] , [Hru87] and more.

The source of pregeometries in model theory is the closure operator onrealizations of a regular type, given by forking. The best known exampleof this, is the algebraic closure operator on a strongly minimal set. Duringthe late 1970s and early 1980s Zilber’s Trichotomy conjecture, which in asomewhat wider context than its original formulation could be viewed as anattempted classification of all such geometries in (super)-stable theories, wasa major driving force in model theory. Zilber conjectured that, after namingsome parameters, the geometry of a strongly minimal set is isomorphic to

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that of a set, that of a vector space or that of an algebraically closed field1.In the mid 1980s Hrushovski [Hru93, Hru92] provided a technique for con-structing new geometries, refuting Zilber’s conjecture. The many possiblevariants allowed by Hrushovski’s constructions still resist any (conjectural)classification in the spirit of Zilber’s Trichotomy.

Whereas Zilber’s conjecture suggested an algebraic dictionary for the clas-sification of geometries (of, say, strongly minimal structures), Hrushovskishows that (many of) his new examples support no algebraic structure atall. In [Hru93], Hrushovski constructs such a strongly minimal structurereferred to as Hrushovski’s ab initio construction. An important step to-wards a reformulation of Zilber’s conjecture is, therefore, a classification ofHrushovski’s examples. In this thesis we provide variants of Hrushovski’sab initio construction, which – we believe – could provide a key to such aclassification.

Hrushovski’s ab initio construction is a Fraısse limit featuring a regulartype whose forking geometry is non-trivial, yet prohibiting the existence ofa definable group. By imposing restrictions on the class from which thelimit is constructed, one produces a strongly minimal structure not fallingwithin Zilber’s conjecture. However, lifting these restrictions on the classproduces an ω-stable limit of Morley rank ω, whose unique type of rank ωis regular and of the same geometrical flavour. The ω-stable version of theconstruction is often referred to as the non-collapsed version in oppositionto the collapsed strongly minimal version. The innovative component of theconstruction, and the one that produces the desired geometry in the limit,is a combinatorial predimension function defined on the amalgamation classused, which determines the dimension function of the non-forking geometryof the limit.

Although we will not consider them in this work, in order to get a clearpicture of the strongly minimal structures we know, Hrushovski fusions mustbe mentioned. In [Hru92] Hrushovski proved that for T1, T2, strongly minimaltheories in disjoint languages L1,L2 with DMP (a minor technical require-ment), there exists a strongly minimal theory T such that its restriction toLi is exactly Ti. The construction itself is technically similar to the ab ini-tio constructions. These fusion constructions further hinder any effort toclassify strongly minimal structures by their geometry. However, in [Hru92]

1 This formulation is not the full conjecture, omitting the existence of definable alge-braic structures in the strongly minimal set.

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Hrushovski suggests that the geometry of the fusion structure is “relativelyflat” over T1 and T2. Naively speaking, algebraic complexity in the fusionmust stem from algebraic complexity in the components, and is restricted bythe degree of complexity of the components.

1.2 Motivation

Unlike the “classic” strongly minimal structures from Zilber’s suggested clas-sification, which are encountered in all of mathematics, Hrushovski’s con-structions remain without an obvious classical template. In the “classic”non-trivial cases, we know of strong ties between a strongly minimal struc-ture’s geometry and what it must interpret. In the case of a modular geome-try, Hrushovski proved in [Hru87] that the structure interprets a vector spaceover a prime field. In the case of a reduct of a curve over an algebraicallyclosed field, by recent work of Hasson and Sustretov [HS17] succeeding workby Rabinovich [Rab93], if the reduct’s geometry is not locally modular, thenit interprets an infinite field. We are still in the dark in the case of the abinitio constructions, but would like to find out whether there is some ana-logue. Perhaps, the geometry exercises some control over what the structuremust interpret.

With the hope of shedding some light on what would be reasonable toconjecture, we study reducts of ab initio constructions. As the collapsed andnon-collapsed versions of the ab initio structures are similar in constructionand geometry [EF12], and the non-collapsed version is significantly better be-haved, it has become common practice to ask questions and look for answersin the non-collapsed construction.

Another reason for examining reducts of ab initio constructions is that thesimilarly constructed Hrushovski fusion constructions, as described above,are characterized by the fact that each of the components of the fusion is areduct of the outcome structure. Although we do not expect such a thing,some of the methods produced in order to tackle reduction in ab initio con-structions may be transferable to fusions. Such a development may assist inrecognizing a structure as a fusion by its model theoretic properties, ratherthan some a-priori knowledge. A recognition of fusions, in turn, immediatelyrekindles a Zilber-like classification of non-fusion strongly minimal geometrytypes. To the author’s best knowledge, model theoretic criteria assertingthat a structure is a fusion construction are yet to be discovered.

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1.3 Overview

The thesis begins with a fully detailed introduction to combinatorial geome-tries (Subsection 1.4.2) and Fraısse-Hrushovski amalgamation classes (Chap-ter 2). Albeit most components of this introduction are standard, sections2.5 and 2.6 are noteworthy. Section 2.5 is the toolbox we use to prove iso-morphism of geometries associated to Fraısse-Hrushovski classes, roughly asin [EF11], [EF12]. Section 2.6 provides explicit means for showing that oneFraısse-Hrushovski limit is the (proper) reduct of another Fraısse-Hrushovskilimit, where both are of the “non-collapsed nature” which we call simple(Definitions 2.3.1, 2.3.3)

In order to properly state the results of Chapter 3, we recall that Hrushovskishowed that if L is a countable (usually finite) relational language, and fora finite L-structure A we let δ(A) := |A| − r(A), where r(A) is the numberof L-relations in (powers of) A, then the class C of all finite L-structuresA such that δ(B) ≥ 0 for all B ⊆ A is an amalgamation class with respectto a class of, so called, self-sufficient (or strong) embeddings. The C-genericstructure M is ω-stable with a unique non-trivial regular type, pC. We call(the pregeometry of) pC the pregeometry of M.

Hrushovski’s ab initio construction from [Hru93] is in the language of aternary relation. In Section 3.1 we present the non-collapsed variation of theconstruction, with varying degrees of symmetry and of arbitrary arity: fixingarity n ≥ 3, for every g ≤ Sn we construct a structureMg and distinguish twoof these – Hrushovski’s construction not admitting any inherent symmetryM 6∼, and Hrushovski’s construction admitting full symmetryM∼. The sectionis concluded with

Theorem (Rephrased 3.1.17). The pregeometry associated to M 6∼ is isomor-phic to the pregeometry associated to Mg, for every g ≤ Sn.

This proves that, for arity n, the level of symmetry has no effect onthe geometry of the structure. Note that, as proved by Evans and Ferreirain [EF11], the geometry does change with n. Section 3.2 is technical, usingthe tools developed in Section 2.6, and culminating in the main result of thechapter:

Corollary (3.2.32). For any g, h ≤ Sn, the structure Mg is isomorphic to aproper reduct of Mh.

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Aside from the ability to choose the level of symmetry with which we’dlike to work later on (when exploring reduction relations with respect toother structures), an immediate consequence of the theorem is that, for ar-ity strictly greater than 2, there is an infinite descending chain of properreducts with non-trivial geometries, beginning with M 6∼. From this, we getthe (weaker) result that there is a strictly ascending chain of closed subgroupsof Sω beginning with the automorphism group of any Mg.

When considering reformulations of Zilber’s classification, before attempt-ing to classify pregeometries of reducts of fusions, a classification of the pre-geometries of reducts of ab initio structures must take place. The reduct〈M 6∼, ϕS(M 6∼)〉 of Section 4.2 is an “obvious counter example” to the mostnaive conjecture, suggesting that every reduct of Hrushovski’s ab initio struc-ture is itself an ab initio structure, referring to “ab initio structures” herein the narrow sense of relational structures that, hereditarily, have at leastas many points as they do relations. In this reduct, the ratio of relationsto points is unbounded when taking into account all finite substructures. InChapter 4, by slightly modifying the construction, we successfully handlethis “counter example” within the scope of the conjecture, arguing that theresulting structure is also an “ab initio” constructions of sorts. Of course,the conjecture can truly be evaluated only after precisely determining whatconstitutes an ab initio structure. There is no widely agreed upon definitionas of yet.

In Section 4.1 we show that Hrushovski’s construction can be carried outin a similar way if, in the context of a unique n-ary relation, instead of defin-ing δ(A) := |A| − r(A) we let δ(A) := |A| −

∑K(|K| − n + 1) where the

sum ranges over the all maximal cliques in A, thus allowing A to support auniformly bounded polynomial (rather than linear) number of relations. Wedub this construction Mclq, arising from the Fraısse-Hrushovski amalgama-tion class Cclq, the clique construction. The structure M∼ can be seen as adegenerate case of the clique construction.

In sections 4.2 and 4.3 we show that Mclq is isomorphic to a (proper)reduct of M 6∼, and that the pregeometries of both structures are isomorphic.In order to simplify the technical proofs of Section 4.1, the class Cclq is slightlylimited in generality. In Section 4.4 we portray the construction of Mclq ingreater generality, using a two-sorted point-line incidence relation, where thelines are imaginary elements corresponding to cliques.

Finally, we conclude the first part of the thesis by defining the geometric

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construction, a variant of the clique construction. The class Cgeo is the classof substructures capturing the finite sub-pregeometries of the pregeometryof M∼. We show that Mgeo, the construction resulting from Cgeo with thepredimension λ, captures the pregeometry of M∼, and is a reduct of thepregeometry of M∼.

1.4 Preliminaries

1.4.1 Notation and conventions

The following are several conventions we use throughout the first four sectionsof the text.

Sets and tuples

• Unless otherwise specified, the letters A,B,C,D . . . will denote finitesets and the letters M,N,X, Y, . . . will denote possibly infinite sets.

• We denote by P(X) the set of subsets of X and by Fin(X) the set offinite subsets of X.

• Let X, Y be sets (possibly infinite) and c, d points. We may, when thereis no confusion, omit the union operator and set notation and denoteXc = X ∪{c}, cY d = Y ∪{c, d}, XY c = X ∪Y ∪{c}, cd = {c, d}, andso on.

• For tuples a = (a1, . . . , ak), b = (b1, . . . , br) we write ab for the concate-nation of the tuples (a1, . . . , ak, b1, . . . , br). Similarly, for an element cdenote ca = (c, a1, . . . , ak) and ac = (a1, . . . , ak, c).

• For a tuple a and a set A, we write a ∈ A to mean that a ∈ A|a|.

First order structures

• Languages are always assumed to be relational and countable. Struc-tures are also always assumed to be countable.

• Calligraphic capital letters (sayM) denote first order structures, withthe respective capital letter denoting the universe of the structure.

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• LetM be a first order structure in a relational language L. We defineits age age(M) to be the collection of all finite L-structures which areisomorphic to some substructure of M.

• For a structure M with universe M , since we assume languages to berelational, a subset N ⊆M may be identified with N , the unique sub-structure ofM with universe N . Write NM to mean this substructureN .

Functions defined on structures

• Let M be a structure in a language L and let R ∈ L be some n-aryrelation symbol. We write R(M) to mean the interpretation of R inM, instead of the more traditional RM. Explicitly, R(M) = {a ∈Mn :M |= R(a)}.We introduce some shorthand:

– If N ⊆M we write R(M/N ) to mean R(M) \R(N ).

– If N ⊆M we write RM(N) to mean R(NM).

– If P ⊆ N ⊆M , we write RM(N/P ) to mean R(NM/PM).

• If A is a finite structure, we denote |A| = |A|.

• Let M be a structure. For A ∈ Fin(M) and a function δ defined onage(M), we write δM(A) to mean δ(A) where A = AM.

• LetM be a structure with a function δ defined on age(M). Let A,B ∈Fin(M), then we denote δM(B/A) = δM(B ∪ A)− δM(A).

Expansions and reducts

• Let L1 ⊆ L2 and letM be an L1-structure. We say that an L2-structureM′ is an expansion ofM if it has universe M and R(M′) = R(M) forall R ∈ L1.

SayM′ is a definable expansion ofM if for each R ∈ L2 there is an L1-formula (with parameters) ϕR(x) such that M′ |= ∀x(R(x)↔ ϕR(x)).

• We define a reduct as follows: An L1-structure M is a reduct of anL2-structure M′, if there exists an L1 ∪ L2-structure M′′, a definable

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expansion of M′ that is also an expansion of M. Note that such astructureM′′ is unique. We sayM is a proper reduct ofM′, ifM′′ isnot a definable expansion of M.

If L1 ⊆ L2 then M is uniquely determined by M′ and L1 and so wewrite M′ � L1 to mean M, and refer to M as the reduct of M′ to L1.

1.4.2 Pregeometries

The contents of this subsection are folklore.

Basic definitions

Let X be a set and let cl : P(X) → P(X). We say (X, cl) is a pregeometryif for all Y, Z ⊆ X:

1. Y ⊆ cl(Y ).

2. If Y ⊆ Z then cl(Y ) ⊆ cl(Z).

3. cl(cl(Y )) = cl(Y ).

4. (Finite nature) If a ∈ cl(Y ) then there is some finite Y0 ⊆ Y such thata ∈ cl(Y0).

5. (Exchange principle) If a ∈ cl(Y b) \ cl(Y ) then b ∈ cl(Y a).

We say Y ⊆ X is independent if a /∈ cl(Y \ {a}) for any a ∈ Y . For Z ⊆ Xwe say Y ⊆ Z is a basis for Z if Y is independent and Z ⊆ cl(Y ).Any Z ⊆ X has a basis and it follows from the exchange principle that if Y1

and Y2 are bases for Z, then |Y1| = |Y2|.For Z ⊆ X we define dim(Z), the dimension of Z to be the cardinality of abasis for Z.

Proposition 1.4.1.

i. A pregeometry (X, cl) is uniquely determined by the collection of finiteindependent subsets of X

ii. A pregeometry (X, cl) is uniquely determined by the restriction of itsdimension function to Fin(X).

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Proof. i. Let Z ⊆ X and let Y be a basis for Z. Then x ∈ cl(Z) if andonly if x ∈ cl(Y ) if and only if there exists some finite Y0 ⊆ Y such thatx ∈ cl(Y0). Thus, x ∈ cl(Z) if and only if there is a finite set Y0 ⊆ Zsuch that Y0 is independent but Y0 ∪ {x} is not independent.

ii. A finite set Y is independent if and only if |Y | = dim(Y ).

It will be useful to us to think of pregeometries as first order structures.Let LPG = {In, Dn}n∈N. To every pregeometry (X, cl) we associate the LPG-structure with universe X, interpreting Ii as the set of independent tuples ofXn and Di as the set of dependent tuples of Xn.

We say that pregeometries (X1, cl1) and (X2, cl2) are isomorphic if theyare isomorphic as first order structures.

Defining a pregeometry from a function

Definition 1.4.2. Let d : Fin(X)→ N. We say that d defines a pregeometryon X if for all Y, Z ∈ Fin(X) and a ∈ X:

I. d(∅) = 0.

II. d(Y ) ≤ d(Y a) ≤ d(Y ) + 1.

III. (Submodularity) d(Y ∪ Z) ≤ d(Y ) + d(Z)− d(Y ∩ Z).

Proposition 1.4.3. Let d : Fin(X)→ N be such that it defines a pregeome-try on X. Then there exists a unique pregeometry (X, cld) whose dimensionfunction restricted to Fin(X) is d. For any Y ∈ Fin(X) the operator cld isdefined

cld(Y ) = {a ∈ X | d(Y a) = d(Y )}

Proof. We define an operator cld : P(X) → P(X) and prove that it inducesa pregeometry on X.For Y ⊆ X finite define:

cld(Y ) = {a ∈ X | d(Y a) = d(Y )}

If Z is finite and Y ⊆ Z such that a ∈ cld(Y ), then by submodularity d(Za) ≤d(Z) +d(Y a)−d(Y ) = d(Z). Since also d(Z) ≤ d(Za), we have equality and

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therefore a ∈ cld(Z). So for finite Z ⊆ X we have cld(Y ) ⊆ cld(Z) for anyY ⊆ Z. Thus, we may extend our definition of cld to any Y ⊆ X:

cld(Y ) =⋃

Y0∈Fin(Y )

cld(Y0)

We prove that (X, cl) satisfies properties 1− 5:

1. For every finite Y0 ⊆ Y we have Y0 ⊆ cld(Y0) ⊆ cld(Y ) and so Y ⊆cld(Y ).

2. Assume Y ⊆ Z. Let a ∈ cld(Y ), then there is some Y0 ∈ Fin(Y ) witha ∈ cld(Y0). Since Y ⊆ Z, Y0 ∈ Fin(Z) and so a ∈ cld(Y0) ⊆ cld(Z). Socld(Y ) ⊆ cld(Z).

3. By 1 we know Y ⊆ cld(Y ) and so by 2, cld(Y ) ⊆ cld(cld(Y )). Now, as-sume a ∈ cld(cld(Y )), then there is some finite set A0 = {a1, . . . , an} inFin(cld(Y )) with a ∈ cld(A0). Since A0 ⊆ cld(Y ), for each i ∈ {1, . . . , n}there is some Y i

0 ∈ Fin(Y ) with ai ∈ cld(Yi

0 ), meaning d(Y i0ai) = d(Y i

0 ).Denote Y0 =

⋃ni=1 Y

i0 . We assume for clarity that ai /∈ Y0 for all

i ∈ {1, . . . , n} and note that this is no loss of generality for the purposeof the following calculation. We calculate by iterating submodularity:

d(Y0A0) =

d(Y0a1 . . . an) ≤ d(Y0a1 . . . an−1) + d(Y n0 an)− d(Y n

0 ) =

d(Y0a1 . . . an−1) ≤ d(Y0a1 . . . an−2) + d(Y n−10 an−1)− d(Y n−1

0 ) =

. . .

d(Y0a1) ≤ d(Y0) + d(Y 10 a1)− d(Y 1

0 ) = d(Y0)

So d(Y0A0) ≤ d(Y0) and therefore d(Y0A0) = d(Y0) by monotonicity ofd (by II). As we also know d(A0a) = d(A0), we may use submodularityonce more, assuming a /∈ Y0 for clarity as before, and receive

d(Y0A0a) ≤ d(Y0A0) + d(A0a)− d(A0)

= d(Y0A0) = d(Y0)

By the above and the monotonicity of d we have d(Y0A0a) = d(Y0).Another application of monotonicity yields d(Y0a) ≤ d(Y0A0a) and sofinally

d(Y0a) ≤ d(Y0A0a) = d(Y0)

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and thus d(Y0a) = d(Y0). Therefore, by definition, a ∈ cld(Y0) ⊆cld(Y ), and so cld(cld(Y )) ⊆ cld(Y ).

In conclusion cld(Y ) = cld(cld(Y )).

4. Immediate from the definition of cld.

5. Assume a ∈ cld(Y b) \ cld(Y ). If b ∈ cld(Y ) then a ∈ cld(cld(Y )) =cld(Y ) in contradiction to our choice of a, so this cannot happen. Sincea ∈ cld(Y b), there is some Y0 ∈ Fin(Y ) such that a ∈ cld(Y0b), sod(Y0ab) = d(Y0b). In addition, a, b /∈ cld(Y0) and in particular d(Y0a) =d(Y0b) = d(Y0) + 1 (by II). Thus, d(Y0ab) = d(Y0b) = d(Y0a) and sob ∈ cld(Y0a) ⊆ cld(Y a).

We have proved that (X, cl) is indeed a pregeometry, denote its dimensionfunction by dim. To conclude, we must prove that for all Y ∈ Fin(X) wehave dim(Y ) = d(Y ).We first show by induction that this is true if Y is independent. Let dim(Y ) =|Y | = n+1. Choose some a ∈ Y , then Y \{a} is independent with |Y \{a}| =n and so by the induction hypothesis d(Y \ {a}) = n. By the fact Y isindependent we have a /∈ cld(Y \ {a}) and so d(Y ) = d(Y \ {a}) + 1 = n+ 1.Now, let Y ∈ Fin(X) be general. Fix an enumeration Y = {a1, . . . , an} anddenote A0 = ∅, Ai+1 = Ai ∪ {ai+1}. Fix Y0 ⊆ Y , a basis of Y , then forany i ∈ {1, . . . , n} we have ai ∈ cld(Y0). By monotonicity of the closure(property 2), this gives us ai+1 ∈ cld(Y0Ai), meaning d(Y0Ai+1) = d(Y0Ai),for any i ∈ {0, . . . , n− 1}. Thus, d(Y ) = d(Y0An) = d(Y0A0) = d(Y0). SinceY0 is independent and is a basis of Y , we have d(Y0) = |Y0| = dim(Y ).

The associated geometry

We say (X, cl) is a geometry if (X, cl) is a pregeometry and in addition:

1. cl(∅) = ∅.

2. cl({x}) = {x} for any x ∈ X.

If (X, cl) is a pregeometry, then it has an associated geometry: Define arelation ∼ on X0 = X \ cl(∅) so that a ∼ b if and only if b ∈ cl({a}). Bythe exchange principle and closure properties of (X, cl), the relation ∼ is an

equivalence relation. Define X = X0/ ∼ and for Y ⊆ X define cl∼(Y ) =

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{[x]∼ ∈ X | x ∈ cl(⋃Y )}. It is easy to verify that this defines a geometry.

The geometry (X, cl∼) is called the geometry associated with (X, cl).

In addition, the dimension function dim∼ of the associated geometry, is givenby the pregeometry’s dimension function dim, and vice versa.For Y ⊆ X:

dim∼(Y ) = dim(⋃

Y )

For Y ⊆ X:dim(Y ) = dim∼({[y]∼ | y ∈ Y \ cl(∅)})

Observation 1.4.4. If (X, clX) and (Y, clY ) are pregeometries such that alltheir equivalence classes under the relation ∼ are of the same cardinality (forexample countably infinite) and | clX(∅)| = | clY (∅)| then they are isomorphicif and only if their associated geometries are isomorphic.

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Chapter 2

Fraısse-Hrushovskiamalgamation classes

Unlike in the case of a Fraısse construction, there is no precise definitionfor a Fraısse-Hrushovski amalgamation class and its associated construction.The notion of a Fraısse-Hrushovski amalgamation class introduced here isgiven in a simple setting which suits the purpose of this work, and may begeneralized in several ways.

Fix L, a countable relational language and fix C0, a class of finite L-structuressuch that:

1. If A ∈ C0 and A ∼= B then B ∈ C0.

2. If A ∈ C0 and B ⊆ A then B ∈ C0

3. The structures of C0 have at most countably many isomorphism types.(i.e: There is a countable family {Ai}∞i=1 ⊆ C0 such that for anyA ∈ C0

there exists some i with A ∼= Ai)

2.1 Predimension

Definition 2.1.1. We say δ : C0 → Z is a predimension function on C0 if ithas the following properties:

1. δ(∅) = 0.

2. If A ∈ C0 with universe A then δ(A) ≤ |A|.

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3. If A ∈ C0 and A ∼= B then δ(A) = δ(B).

4. If D ∈ C0 and A,B ⊆ D then δD(A∪B) ≤ δD(A)+δD(B)−δD(A∩B).(Submodularity)

Fix δ : C0 → Z, a predimension function on C0. For anyM with age(M) ⊆C0 we define a function dM : Fin(M)→ Z∪{−∞} such that for A ∈ Fin(M)

dM(A) = min {δM(B) | B ∈ Fin(M), A ⊆ B ⊆M}

We call dM the dimension function associated with δM.Let C be the subclass of C0 whose elements are all the structures A ∈ C0

such that dA(∅) = 0, or synonymously, the structures for which dA is non-negative. Take note that C also is closed under isomorphism and undertaking sub-structures, and that there are only countably many isomorphismtypes in C.

Proposition 2.1.2. Let M be an L-structure with age(M) ⊆ C. Then thefunction dM : Fin(M)→ N defines a pregeometry on M .

Proof. We only need to show that dM has the properties specified in Defini-tion 1.4.2.

I. Clearly dM(∅) ≥ 0 and also dM(∅) ≤ δM(∅) ≤ |∅| = 0. So dM(∅) = 0.

II. Let A ∈ Fin(M) and a ∈ M . By definition, dM(A) ≤ dM(Aa). Now,choose some B ∈ Fin(M) with A ⊆ B and δM(B) = dM(A). Bydefinition, since Aa ⊆ Ba we have dM(Aa) ≤ δM(Ba). If a /∈ B thenδM(Ba) ≤ δM(B) + δM(a) ≤ dM(A) + 1, and otherwise δM(Ba) =dM(A). In any case, this gives us dM(Aa) ≤ dM(A) + 1.

III. Let A,B ∈ Fin(M). Choose some A ⊆ A ∈ Fin(M) such that δM(A) =dM(A) and B ⊆ B ∈ Fin(M) such that δM(B) = dM(B). ThenA ∪ B ⊆ A ∪ B and in particular dM(A ∪ B) ≤ δM(A ∪ B), and alsoA ∩B ⊆ A ∩ B, so dM(A ∩B) ≤ δM(A ∩ B). Thus we have

dM(A ∪B) ≤ δM(A ∪ B)

≤ δM(A) + δM(B)− δM(A ∩ B)

≤ dM(A) + dM(B)− dM(A ∩B)

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For M with age(M) ⊆ C0 and N ∈ P(M), we say that the d-closure ofN in M is the set

clM(N) =⋃

A∈Fin(N)

{a ∈M | dM(Aa) = dM(A)}

We say that N is d-closed inM if N equals its d-closure inM. If age(M) ⊆C as in the above proposition, the d-closure inM of a set N ∈ P(M) is justthe closure of N in the pregeometry defined by dM.

Notation. The pregeometry arising from an L-structureM is denoted PG(M).

2.2 Self-sufficiency

In this section we denote by M a general L-structure with age(M) ⊆ C0.

For a finite substructure A ⊆ M we say that A is self-sufficient in M anddenote A 6M if dM(A) = δM(A).For N ∈ P(M) and A ∈ Fin(N) we write A 6M N to mean A 6 N whereA = AM and N = NM. We abuse notation and write A 6M for A 6M M .It is important to note that for us A 6 M implies A ⊆ M. In addition,A 6M N implies that A,N ⊆M .

Notation. Recall the notation δM(B/A) = δM(A∪B)− δM(A). Extend itto the case of an infinite set X ⊆M replacing A, by defining

δM(B/X) = min {δM(B/A) | A ∈ Fin(X), B ∩X ⊆ A}.

By submodularity, had X been finite, the value of the above right-hand sidewould coincide with δM(B/X).Whenever X ⊆ N ⊆M with (N \X) finite, we write δM(N/X) to implicitlymean δM((N \X)/X).

We may use the notation above to restate the submodularity of δM:

• For any X1, X2 ∈ Fin(M) with X1 ∩ X2 = Y we have δM(X2/X1) ≤δM(X2/Y ).

Observe that the statement remains true for arbitrary X1, X2, as long as(X2 \X1) is finite.Similarly, we restate the definition for self-sufficiency, accommodating infinitesets:

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• For X ∈ P(M) and N ∈ P(M), X 6M N if and only if X ⊆ N andδM(B/X) ≥ 0 for all B ∈ Fin(N).

Lemma 2.2.1. N 6 M if and only if δM(B/B ∩ N) ≥ 0 for any B ∈Fin(M).

Proof. Assume N 6 M and let B ∈ Fin(M). By submodularity we haveδM(B/N) ≤ δM(B/B∩N). By N 6M we have δM(B/N) ≥ 0. Combiningthe two gives δM(B/B ∩N) ≥ 0.

Assume N 66M. Then there are C ∈ Fin(M) and C ∩N ⊆ A ⊆ N finitesuch that δM(C/A) < 0. Denoting B = A∪C, we have that δM(B/B∩N) =δM(C/A) < 0.

Proposition 2.2.2. The relation 6 is transitive.

Proof. Let X 6 N and N 6M. Let C ∈ Fin(M) and let A ∈ Fin(X) besuch that C ∩X ⊆ A. Denote B = (C ∩ N) ∪ A. Then δM(C/B) ≥ 0 andδM(B/A) ≥ 0 by N 6 M and X 6 N respectively. Thus, δM(C/A) ≥ 0.Since C and A were arbitrary, X 6M.

Lemma 2.2.3. If X 6M and Y 6M then X ∩ Y 6M.

Proof. For any B ∈ Fin(X), by Y 6M and Lemma 2.2.1 we get δM(B/B ∩Y ) ≥ 0. So, by Lemma 2.2.1 we have X ∩ Y 6M X. Together with X 6Mand the transitivity of 6, this gives us X ∩ Y 6M.

Assume for a moment that −∞ /∈ Im(dM). For any A ∈ Fin(M), ifA ∈ Fin(M) with δM(A) = dM(A) and A ⊆ A, then A 6M. Lemma 2.2.3guarantees that the intersection of all such sets A for a given A ∈ Fin(M)is also self-sufficient in M. Thus, we may unambiguously speak of the self-sufficient closure of A inM, the smallest superset of A which is self-sufficientin M . It is easy to see that the self-sufficient closure of a finite set is finite.

Lemma 2.2.4. If N 6 M then dM � Fin(N) = dN . Addditionally, if−∞ /∈ Im(dN ) then the converse holds.

Proof. Let Y ∈ Fin(N). Let X ∈ Fin(M) be such that Y ⊆ X. By Lemma2.2.1, δM(X ∩ N) ≤ δM(X). Furthermore, δN (X ∩ N) = δM(X ∩ N) andY ⊆ X ∩ N , so dN (Y ) ≤ δM(X). Since X was arbitrary, we get dN (Y ) ≤dM(Y ). As by definition dM(Y ) ≤ dN (Y ), we have dM(Y ) = dN (Y ).

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Now assume −∞ /∈ Im(dN ) and N 66M. By Lemma 2.2.1 there is someB ∈ Fin(M) with δM(B/B ∩ N) < 0. Let A ∈ Fin(N) be such that A ⊇(B ∩ N) and δM(A) = dN (B ∩ N), and note that dN (A) = δN (A). Bysubmodularity we have δM(B/A) < 0, and therefore dN (A) 6= dM(A).

Lemma 2.2.5. If N is d-closed in M then N is self-sufficient in M.

Proof. Assume N is d-closed in M and denote N = NM. Contrary to thestatement, assume N 66M. Then there is some B ∈ Fin(M) with δM(B/B∩N) < 0. Denote A = B ∩ N and let b ∈ B. For any arbitrary A ⊆ C ⊆ Nwe have by submodularity δM(BC/C) < 0 and thus dM(Ab) ≤ δM(BC) <δM(C). As d-closedness implies dM(A) = dN (A), we may take C with δM(C)arbitrarily close to dM(A) which gives us dM(Ab) ≤ dM(A). By definitionwe have dM(A) ≤ dM(Ab), and so dM(A) = dM(Ab) which places b ∈ N . SoB ⊆ N in contradiction to δM(B/B ∩N) < 0.

Lemma 2.2.6. If X is d-closed in M and N 6M, then N ∩X is d-closedin N .

Proof. By Lemma 2.2.4, dM � Fin(N) = dN . Denote Y = N ∩ X and leta ∈ N be such that dN (Y a) = dN (Y ). By submodularity, dM(a/X) ≤dM(a/Y ) = dN (a/Y ) = 0. As dM(a/X) ≥ 0 by definiton of the dimensionfunction, we get dM(Xa) = dM(X) and so a ∈ X, by d-closeness of X. Thus,a ∈ Y and so Y is d-closed in N .

For an L-structureN and an embedding ofN intoM (in the sense of firstorder structures) f : N →M, we say f is a strong embedding if f [N ] 6M.

Corollary 2.2.7. The class of strong embeddings is closed under compositionand contains all isomorphism maps.

2.3 Amalgamation

We say that (C, δ) is a Fraısse-Hrushovski amalgamation class if wheneverA,B1,B2 ∈ C with strong embeddings fi : A → Bi (for i ∈ {1, 2}), thereexists some D ∈ C and strong embeddings gi : Bi → D such that g1 ◦ f1 =g2 ◦ f2.

Definition 2.3.1. Let A,B1,B2 ∈ C0 such that A ⊆ Bi for i ∈ {1, 2} andB1 ∩ B2 = A. We say D is a simple amalgam of B1 and B2 over A (withrespect to δ) if it has the following properties:

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1. D ∈ C0.

2. The universe of D is D = B1 ∪B2

3. The induced substructures of D on the sets A,B1, B2 are A,B1,B2

respectively.

4. For all A ⊆ X ⊆ D, the equality δD(X/A) = δD(X ∩ B1/A) + δD(X ∩B2/A) holds.

Remark. In the above definition, A = ∅ is allowed.

Observation. Let D be a simple amalgam of B1 and B2 over A. Let A ⊆E ⊆ D and denote Ei = E ∩ Bi. Then E is a simple amalgam of E1 and E2

over A.

Lemma 2.3.2. Let D be a simple amalgam of B1 and B2 over A. Then:

1. If A 6 B1 then B2 6 D and D ∈ C.

2. If in addition A 6 B2, then A 6 D.

Proof. Let B2 ⊆ X ⊆ D. Then:

δD(X/B2) = δD(X)− δD(B2)

= δD(X/A)− δD(B2/A)

= δD(X ∩B1/A) + δD(X ∩B2/A)− δD(B2/A)

= δD(X ∩B1/A)

Now assume that A 6 B1, then for any such X, we have δD(X/B2) ≥ 0 andtherefore B2 6 D. By transitivity of 6 and ∅ 6 B2 we have ∅ 6 D andtherefore D ∈ C.Assume additionally that A 6 B2, then by transitivity and B2 6 D we haveA 6 D.

Definition 2.3.3. If for all A,B1,B2 ∈ C0 with A 6 B1, A ⊆ B2, andB1 ∩ B2 = A there exists a simple amalgam of B1 and B2 over A , we saythat (C, δ) is a simple Fraısse-Hrushovski amalgamation class.

Lemma 2.3.4. If (C, δ) is a simple Fraısse-Hrushovski amalgamation class,then in particular it is a Fraısse-Hrushovski amalgamation class.

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Proof. Let A,B1,B2 ∈ C be structures with strong embeddings fi : A → Bi.Without loss of generality, by renaming elements (this is no loss of generalityby Corollary 2.2.7) we may assume f1, f2 are identity maps and B1∩B2 = A.Take D to be a simple amalgam of B1 and B2 over A and let g1, g2 be theidentity maps on B1 and B2 respectively. Because A 6 Bi for i ∈ {1, 2}, bythe previous lemma, D ∈ C and g1, g2 are strong embeddings. So D fulfilsthe requirements.

In a simple Fraısse-Hrushovski amalgamation class, we may take a simpleamalgam over A of any finite list of structures A 6 B1, . . . ,Bn with Bi∩Bj =A by inductively amalgamating them one by one. Define D0 = A and defineDi+1 to be a simple amalgam of Di and Bi+1 over A. By construction, theresulting structure D = Dn satisfies the following analogous properties:

1. D ∈ C0.

2. The universe of D is D =⋃ni=1 Bi

3. The induced substructures ofD on the setsA,B1, . . . , Bn areA,B1, . . . ,Bnrespectively.

4. For all A ⊆ X ⊆ D, the equality δD(X/A) =∑n

i=1 δD(X∩Bi/A) holds.

It is worth noting that for a class to always have a simple amalgam isstrictly stronger than being a Fraısse-Hrushovski amalgamation class. Forexample, the class used in the original construction by Hrushovski in [Hru93](denoted in Hrushovski’s paper by C0) did not possess this property.

2.4 The generic structure

Definition 2.4.1. We say that a countable structure M is a generic struc-ture for (C, δ) if age(M) ⊆ C and M has the following property:

(∗) If A 6 M and A 6 B for B ∈ C, then there is a strong embeddingf : B →M such that f � A is the identity map on A.

Note that if M is a generic structure for (C, δ), then age(M) = C, sinceevery structure in C may be strongly embedded into M over the empty set.

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Lemma 2.4.2. Let M and N be generic structures for (C, δ). Let f0 : A→B be a finite partial isomorphism between M and N with A 6M M andB 6N N . Then for any c ∈M , there exists some f : C → D, a finite partialisomorphism between M and N extending f0 such that:

1. A ⊆ C 6M

2. B ⊆ D 6 N

3. c ∈ C

Proof. Without loss of generality, rename the elements of B so that f0 is theidentity map. Let C be the self-sufficient closure of Ac in M, then C 6Mand a ∈ C. Denote C = CM and A = AM = AN .

Since A ⊆ C ⊆ M and A 6 M, by definition A 6 C. So A 6 N ,A 6 C and C ∈ C. Since N is a generic structure for (C, δ), there is a strongembedding f : C → N such that f � A is the identity map, and in particularf � A = f0. Taking D = Im(f) concludes the proof.

Corollary 2.4.3. Let M, N be generic structures for (C, δ).

i. If f0 : A → B is a finite partial isomorphism with A 6M and B 6 N ,then f0 extends to an isomorphism of structures f :M→N .

ii. M∼= N

Proof. i. By back and forth between self-sufficient substructures using theabove lemma, since both M and N are countable.

ii. By the previous item, taking f0 to be the empty map.

Proposition 2.4.4. If (C, δ) is a Fraısse-Hrushovski amalgamation classthen there exists a unique (up to isomorphism) generic structure for (C, δ).

Proof. By the above corollary, we need only prove existence.Let M be a countable infinite set. Let A ⊆ C be a countable family ofstructures, closed under taking substructures, such that for any B ∈ Cthere exists some A ∈ A with B ∼= A. Consider the family of maps F ={f : A→M | A ∈ A} and note that it is countable. Consider the family oftuples T = {(f,A,B) | B ∈ A, A 6 B, f ∈ F, Dom(f) = A} and note thatit is countable since T ⊆ F × A× A.

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Let {(fi,Ai,Bi)}∞i=0 be an enumeration of T such that each tuple of T isrepeated infinitely many times. We inductively build an ascending chain ofstructures

M0 ⊆M1 ⊆M2 ⊆ . . .

such that Mi ⊆M ,Mi ∈ C andMi 6Mi+1 for every i ∈ N. We arbitrarilychoose M0 = ∅.

Assume thatMi is given. If Im(fi) 66Mi, defineMi+1 =Mi. Otherwise,fi is a strong embedding of A into Mi. Since (C, δ) is a Fraısse-Hrushovskiamalgamation class, there exists some D ∈ C with strong embeddings g1 :Mi → D and g2 : Bi → D such that g1 ◦ fi = g2 � A. We fix such a D. Byrenaming the elements of D, we may assume that g1 is the identity, meaningMi 6 D. Since M \Mi is infinite, we may further rename the elements ofD \Mi so that D ⊆M . Define Mi+1 = D.

We takeM =⋃∞i=0Mi. ThenM is a countable L-structure with age(M) ⊆

C. By transitivity of 6 we haveMi 6M for any i ∈ N. We show that M isa generic structure for (C, δ).

Let A 6 M and let A 6 C for some C ∈ C, then there is some B ∈ Awith C ∼= B. Thus, without loss of generality we may assume C ∈ A. SinceA is closed under taking substructures, we also have A ∈ A. Fix j ∈ Nlarge enough such that A ⊆ Mj and note that since A 6 M, by definitionA 6 Mi for any i ≥ j. By taking f to be the identity map on A, we get(f,A,B) ∈ T and in particular (f,A,B) = (fi,Ai,Bi) for some i > j. Byconstruction, there is a strong embedding g : B →Mi+1 such that f = g � A.So g : C → Mi+1 is a strong embedding such that g is the identity map onA. Since Mi+1 6M, in particular g is a strong embedding into M.

Say (C, δ) is a Fraısse-Hrushovski amalgamation class, we call its uniquegeneric structure M the Fraısse-Hrushovski limit of (C, δ). By Proposition2.1.2, dM defines a pregeometry (M, cld) on M . We call this pregeometrythe pregeometry associated to M or the pregeometry associated to (C, δ), anddenote it PG(M) or PG (C, δ) unambiguously.

2.5 Isomorphism of pregeometries

Fix (C1, δ1) and (C2, δ2), two Fraısse-Hrushovski amalgamation classes withassociated schemes of dimension d1, d2, notions of self-sufficiency 61, 62,and generic structures M1, M2, respectively.

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The following definition and the proposition that immediately follows area slight variant of [EF12, Lemma 3.2].

Definition 2.5.1. We say (C1, δ1) geometrically extends to (C2, δ2) and de-note (C1, δ1) (C2, δ2) if the following holds:

(∗) Let A1 ∈ C1, A2 ∈ C2 be such that PG(A1) ∼= PG(A2) with f0 anisomorphism of pregeometries witnessing this. Then for every C ∈ C1

such that A1 61 C there exist B1 ∈ C1, B2 ∈ C2 such that C 61 B1,A2 62 B2, and some isomorphism of pregeometries f ⊇ f0 witnessingPG(B1) ∼= PG(B2).

Observation 2.5.2. Since the class C2 is closed under isomorphisms, whenproving that (C1, δ1) (C2, δ2), in the notation of the definition above, wemay assume that f0 is the identity map and A1, A2 have the same universeA. We use this observation implicitly in future proofs.

Proposition 2.5.3. If (C1, δ1) (C2, δ2) and (C2, δ2) (C1, δ1) thenPG(M1) ∼= PG(M2).

Proof. We show that there exists an isomorphism f : M1 → M2 of prege-ometries, by a back and forth argument.

For i = 1, 2, let Ai 6i Mi and denote Ai = AMii . Assume that fj :A1 → A2 is a bijection witnessing PG(A1) ∼= PG(A2). Let m ∈M1 be someelement of M1 and let C 61 M1 be the self-sufficient closure of A1m in M1.Denote C = CM1 . Then A1 61 C and C ∈ C1 and so by (C1, δ1) (C2, δ2)there exist some B1 ∈ C1 with C 61 B1 and B2 ∈ C2 with A2 62 B2 andg : B1 → B2, extending fi, witnessing PG(B1) ∼= PG(B2). Since M1 and M2

are generic, we may assume that Bi 6i Mi and take fj+1 = g.Taking f0 = ∅ and alternating between back and forth at each step, we

construct f =⋃j∈N fj.

The following lemma and corollary are technical tools, to be used later,in proofs of geometrical extension between Fraısse-Hrushovski amalgamationclasses.

Lemma 2.5.4. Let B1 ∈ C1 and B2 ∈ C2 be two structures with universeB and let A ⊆ B be such that A 61 B1 and A 62 B2. Denote δi = δi

Bi,di = dBii , and say that a set is di-closed to mean that is d-closed with respectto di. Assume that (d1 � Fin(A)) = (d2 � Fin(A)). Then:

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i. If Y ⊆ B is d1-closed in B1, then δ1(Y ∩ A) = δ2(Y ∩ A).

ii. If Y ⊆ B is d1-closed in B1 and δ1(Y/Y ∩ A) ≥ δ2(Y/Y ∩ A), thend1(X) ≥ d2(X) for any X ⊆ B whose d1-closure in B1 is Y .

iii. If δ1(Y/Y ∩A) = δ2(Y/Y ∩A) for any Y ⊆ B di-closed in Bi, for eitheri = 1 or i = 2, then d1 = d2.

Proof. Denote Ai = ABi . By Ai 6i Bi and Lemma 2.2.4, dAii = (di � Fin(A)).In particular, dA1

1 = dA22 .

i. By Lemma 2.2.6, Y ∩ A is d1-closed in A1. Since dA11 = dA2

2 , it must bethat Y ∩A is also d2-closed in A2. By Lemma 2.2.5, we have Y ∩A 6i Aiand so δ1(Y ∩ A) = dA1

1 (Y ∩ A) = dA22 (Y ∩ A) = δ2(Y ∩ A).

ii. Since δ1(Y/Y ∩A) ≥ δ2(Y/Y ∩A), by the previous item δ1(Y ) ≥ δ2(Y ).Let X ⊆ B be such that its d1-closure in B1 is Y . Then d1(X) = d1(Y ).Since Y is d1-closed in B1, it is also self-sufficient in B1 and therefored1(Y ) = δ1(Y ). So d1(X) ≥ δ2(Y ). As X ⊆ Y , by definition δ2(Y ) ≥d2(X). In conclusion, d1(X) ≥ d2(X).

iii. Immediate by symmetry of B1 and B2 and the previous item.

2.6 Reducts of generic structures

Fix (C1, δ1) and (C2, δ2), two Fraısse-Hrushovski amalgamation classes withassociated notions of self-sufficiency 61, 62, and generic structures M1, M2.We wish to find a sufficient condition for M2 to be isomorphic to a (proper)reduct of M1.

Let L1 be the language of structures in C1 and let L2 be the languageof structures in C2. For each R ∈ L2 fix some L1-formula ϕR(x) (withoutparameters1). For any structure M with age(M) ⊆ C1, let M+ be thedefinable expansion of M to L1 ∪ L2 such that for all R ∈ L2

M+ |= ∀x(R(x)↔ ϕR(x))

1One can allow the use of a finite set of parameters from M1 by restricting the followingdiscussion to structures of C1 containing them.

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and denote M � L2 =M+ � L2. Denote M(2) = M1 � L2.The following definitions are with respect to our choice of {ϕR}R∈L2 , and sois the remainder of this section.

Definition 2.6.1. We say that 61 encloses reducts if whenever A ∈ C1, Mis a structure with age(M) ⊆ C1, and A 61 M, also AM�L2 = (A � L2).

Definition 2.6.2. We say that C1 reduces to C2 if for every A ∈ C1 we have(A � L2) ∈ C2.

Lemma 2.6.3. If 61 encloses reducts, C1 reduces to C2 and age(M) ⊆ C1,then age(M � L2) ⊆ C2.

Proof. Denote M(2) = M � L2. Let A ∈ Fin(M). Let A ∈ age(M) bethe induced structure on the self-sufficient closure of A in M. By A 61 Mwe have AM(2) = (A � L2), and by A ∈ C1 we have (A � L2) ∈ C2. SinceAM(2) ⊆ AM(2) ∈ C2 we also have AM(2) ∈ C2. Thus, age(M(2)) ⊆ C2.

Definition 2.6.4. We say that 61 is stronger than 62 if whenever age(M) ⊆C1 and A 61 M, also A 62 (M � L2).

Lemma 2.6.5. If 61 encloses reducts, C1 reduces to C2, and (C1, δ1) is asimple Fraısse-Hrushovski amalgamation class, then 61 is stronger than 62.

Proof. It is enough to show that for B ∈ C1, if A 61 B then A 62 (B � L2).Denote B(2) = (B � L2), A = AB, and A(2) = AB(2) . Because 61 enclosesreducts, we also have A(2) = (A � L2). Since C1 reduces to C2 we haveA(2),B(2) ∈ C2. As B is arbitrary, it suffices to show that δ2(B(2)/A(2)) ≥ 0.

Since B ∈ C1, we may assume that B 61 M1 and consequently A 61 M1.Let r = δ2(A(2)) + 1, and note r > 0.

Let B1, . . . ,Br be structures in C1 such that Bi ∩ Bj = A for all i < jand Bi ∼= B over A for all i. Because C1 is a simple Fraısse-Hrushovskiamalgamation class, there exists D, a simple amalgam of all the Bi over A.By the properties of a simple amalgam and the fact A 61 Bi for all i, wehave A,B1, . . . ,Br 61 D and D ∈ C1. By A 61 M1, A 61 D, and the fact M1

is a generic structure for (C1, δ1), we may strongly embed D into M1 over A.Without loss of generality, assume D 61 M1.

Since Bi 61 M1 for every 1 ≤ i ≤ r, we have BM(2)

i = (Bi � L2) ∼= B(2).Since A 61 M1 we have AM(2) = (A � L2) = A(2). Denote D(2) = DM(2) andnote that since D 61 M1 we have D(2) = (D � L2) ∈ C2.

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By submodularity,

δ2(D(2)/A(2)) ≤∑

1≤i≤r

δ2(BM(2)

i /A(2)) = r · δ2(B(2)/A(2))

Now

δ2(D(2)) = δ2(D(2)/A(2)) + δ2(A(2))

≤ r · δ2(B(2)/A(2)) + (r − 1)

< r · (δ2(B(2)/A(2)) + 1)

But D(2) ∈ C2 and so δ2(D(2)) ≥ 0. Thus, it must be that δ2(B(2)/A(2)) > −1and the lemma is proven.

Definition 2.6.6. We say that (C1, δ1) has a mixed amalgam for (C2, δ2) iffor every A ∈ C1 and B ∈ C2 such that (A � L2) 62 B there exists someC ∈ C1 such that A 61 C and B 62 (C � L2).

Proposition 2.6.7. Assume that:

• (C1, δ1) is a simple Fraısse-Hrushovski amalgamation class.

• (C2, δ2) is a simple Fraısse-Hrushovski amalgamation class.

• C1 reduces to C2.

• 61 encloses reducts.

• (C1, δ1) has a mixed amalgam for (C2, δ2).

Then M(2)∼= M2.

Proof. By Proposition 2.4.4, it suffices to show thatM(2) is a generic structurefor (C2, δ2). Let A 62 M(2) with universe A and let B ∈ C2 such that A 62 B,we must find a strong embedding of B into M(2) over A.

Let A be the self-sufficient closure of A in M1. Denote A1 = AM1 andA(2) = AM2 . Because 61 encloses reducts, also A(2) = (A1 � L2). Note thatA 62 A(2), and so because (C2, δ2) is a simple Fraısse-Hrushovski amalga-mation class, we may take D to be a simple amalgam of A(2) and B over A.By Lemma 2.3.2, A(2) 62 D, B 62 D and D ∈ C2.

By the fact (C1, δ1) has a mixed amalgam for (C2, δ2), let C ∈ C1 besuch that A1 61 C and D 62 (C � L2). By the property of M1 as a generic

29

structure, we may strongly embed C into M1 over A1. Assume withoutloss of generality that C 61 M1. Because 61 encloses reducts, we haveCM(2) = (C � L2) and so BM(2) = B. Thus, we have found an embedding of Binto M(2) over A.

All of the conditions of Lemma 2.6.5 hold and so 61 is stronger than62. Thus, (C � L2) 62 M(2). By our choice of C and transitivity, alsoB 62 (C � L2), and therefore B 62 M(2). This completes the proof.

Definition 2.6.8. We say that (C1, δ1) is lenient towards (C2, δ2) if for anyF ∈ C1 there are structures A,B ∈ C1 such that

• F 61 A and F 61 B.

• A and B are not isomorphic over F .

• (A � L2) and (B � L2) are isomorphic over F

Lemma 2.6.9. Under the assumptions of the above proposition, assume ad-ditionally that (C1, δ1) is lenient towards (C2, δ2). Then M1 is not a definableexpansion of M(2) over any finite parameter set. In particular, if L2 is finite,M(2) is a proper reduct of M1.

Proof. If M(2) is not a proper reduct of M1 and M+1 is a definable expansion

of M(2), say over the parameter set F ⊆M1, then any automorphism of M(2)

which fixes F pointwise is also an automorphism of M+1 , and in particular

M1. We show that this is not the case when F is finite, using the fact thatM(2) is a generic structure for (C2, δ2).

The above paragraph is true of any parameter set F , so we may assumeF 61 M1 by replacing any parameter set F with its self-sufficient closure inM1. Let A and B be as promised by lenience with respect to F = FM1 . ByF 61 A,B and the fact M1 is generic, A and B may be strongly embeddedinto M1 over F . Without loss of generality, assume A,B 61 M1.

Because 61 encloses reducts we have AM(2) = (A � L2) and BM(2) = (B �L2). By Lemma 2.6.5, 61 is stronger than 62, and so we also have A 62 M(2)

and B 62 M(2). By lenience, let f0 be an isomorphism of AM(2) and BM(2) overF . Thus, f0 is a finite partial isomorphism between strong substructures ofM(2) and so by 2.4.3.i f0 extends to some f , an automorphism of M(2). SinceA and B are not isomorphic over F , the bijection f is not an automorphismof M1.

Since F was general, as explained, M+1 cannot be a definable expansion

of M(2) over a finite parameter set.

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Remark 2.6.10. In fact, the contents of this section can be applied in gen-erality greater than that of reducts – e.g. “reducts” to definable types, ormore intricate maps from L1-structures to L2-structures:

Assume to each L1-structureM with age(M) ⊆ C1, there is an associated

L2-structure M with the same universe as M. Furthermore, assume that ifN ∼= M then N ∼= M (by the same bijection). If the analogue conditions

of 2.6.7 hold, then M1∼= M2. If the analogue conditions of 2.6.9 hold, then

additionally there is an automorphism of M1 which is not an automorphismof M1.

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Chapter 3

Hrushovski’s ab initioconstructions of varyingdegrees of symmetry

3.1 The structures Mg and their pregeome-

tries

In this section we present a class of Fraısse-Hrushovski amalgamation classes,as discussed in Chapter 2. We will use the notation for integer valued func-tions presented in Subsection 1.4.1 with no explicit reference.The contents of subsections 3.1.1 are mostly standard ( [EF11, Section 2],[Zie13, Section 2]).

3.1.1 The class Cg

Fix some n ∈ N. Fix some subgroup g ≤ Sn (Where Sn is the symmetricgroup on n elements). Let Lg = {Rg} where Rg is an n-ary relation.

Definition 3.1.1. Let Cg0 be the class of finite L-structures A such that:

• If A |= Rg(a1, . . . , an) then a1, . . . , an are distinct elements.

• A |= ∀x1, . . . , xn[Rg(x1, . . . , xn)↔∧σ∈gRg(xσ(1), . . . , xσ(n))]

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Note that Cg0 is closed under isomorphism and taking substructures, and

since it is a class of finite structures in a finite language, Cg0 has countably

many isomorphism types.In the context of Cg

0, an ordered tuple of elements is related in Rg if andonly if any permutation of the tuple that lies in g, is also related in Rg. It isthus best to think of orbits under g instead of actual tuples.

Definition 3.1.2. Let x1, . . . , xn be distinct elements. Define:

[x1, . . . , xn]g = {(xσ(1), . . . , xσ(n)) : σ ∈ g}

For a structure M with age(M) ⊆ Cg0 define

Rg[M] = {[a1, . . . , an]g :M |= Rg(a1, . . . , an)}

Remark. In structures of Cg0, we think of relations as tuples of the form

[a1, . . . , an]g rather than ordered tuples (a1, . . . , an). If when defining an Lg-structureM we define Rg[M] instead of Rg(M), it is implied that Rg(M) =⋃Rg[M].

We call an object of the form [x1, . . . , xn]g a g-tuple or a g-relation.

Notation. We use the same notational shorthand for Rg[M] as we do forR(M), i.e.

1. If N ⊆M we write Rg[M/N ] to mean Rg[M] \Rg[N ].

2. If N ⊆M we write RMg [N ] to mean Rg[NM].

3. If P ⊆ N ⊆M , we write RMg [N/P ] to mean Rg[NM/PM].

Also, in the context of formulas, we use shorthand and write Rg[x1, . . . , xn]to mean

∧σ∈gRg(xσ(1), . . . , xσ(n)).

Definition 3.1.3. Define for A ∈ Cg0:

rg(A) = |{[a1, . . . , an]g : A |= Rg(a1, . . . , an)}| = |Rg[A]|δg(A) = |A| − rg(A)

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Observation. If M is an Lg-structure with age(M) ⊆ Cg0 and A,B ∈

Fin(M), then

δMg (B/A) = (|A ∪B| − rMg (A ∪B))− (|A| − rMg (A))

= (|A ∪B| − |A|)− (rMg (A ∪B)− rMg (A))

= |B \ A| − rMg (B/A)

Informally, if we think of B as being “added” to A, this value is the numberof new points added minus the number of new g-relations formed.

Lemma 3.1.4. The function δg is submodular. Explicitly, if D ∈ Cg0 and

A,B ⊆ D, then δDg (A ∪B) ≤ δDg (A) + δDg (B)− δDg (A ∩B).In addition, equality holds if and only if RDg [A ∪B] = RDg [A] ∪RDg [B].

Proof. We have |A ∪B| = |A|+ |B| − |A ∩B| and

rDg (A ∪B) = |RDg [(A ∪B)]|≥ |RDg [A] ∪RDg [B]|= |RDg [A]|+ |RDg [B]| − |RDg [A] ∩RDg [B]|= |RDg [A]|+ |RDg [B]| − |RDg [(A ∩B)]|= rDg (A) + rDg (B)− rDg (A ∩B)

In addition, equality holds between the first and second lines if and only ifRDg [(A ∪B)] = RDg [A] ∪RDg [B].

By subtracting the second inequality from the first, we receive exactlythe statement of the lemma.

Corollary 3.1.5. The function δg is a predimension function on Cg0.

Proof. By its definition, δg is perserved under isomorphisms, δg(A) ≤ |A| forany A ∈ Cg

0, and in particular δg(∅) = 0. Submodularity is proved in thelemma above.

As δg is a predimension function on Cg0, as in Chapter 2, we associate

with it a scheme of dimension functions dMg and a notion of self-sufficiency6g.

Corollary 3.1.6. For any A,B1,B2 ∈ Cg0 such that A ⊆ Bi for i ∈ {1, 2}

and B1 ∩B2 = A, there exists a simple amalgam of B1 and B2 over A (withrespect to δg).

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Proof. Recall the definition of a simple amalgam (Definition 2.3.1). Considerthe structure D with

D = B1 ∪B2

Rg[D] = Rg[B1] ∪Rg[B2]

We show that D is a simple amalgam of B1 and B2 over A. The only non-trivial property we must verify is that δDg (X/A) = δDg (X ∩ B1/A) + δDg (X ∩B2/A) for all A ⊆ X ⊆ D. Since for any A ⊆ X ⊆ D the structure XD isconstructed from (X ∩B1)D and (X ∩B2)D the same way D was constructedfrom B1 and B2, it suffices to show δDg (D/A) = δDg (B1/A) + δDg (B2/A). Thisfollows by construction and the additional part of Lemma 3.1.4.

Definition 3.1.1. Cg = {A ∈ Cg0 | ∅ 6g A}.

By the above corollary and Lemma 2.3.4, (Cg, δg) is a (simple) Fraısse-Hrushovski amalgamation class. We denote the Fraısse-Hrushovski limit of(Cg, δg) by Mg and recall that it has the following defining property:

(∗) If A 6g Mg and A 6g C for C ∈ Cg, then there is a strong embeddingf : C →Mg such that f � A is the identity map on A.

Notation 3.1.7. The cases where g is trivial or all of Sn will be of partic-ular interest to us. For the case g = {1Sn} we replace (Lg, Rg,Cg

0,Cg, δg, dg,6g

,Mg) with the notation (L 6∼, R6∼,C 6∼0 ,C 6∼, δ6∼, d6∼,66∼,M 6∼) and denote (a1, . . . , an)instead of [a1, . . . , an]g.Similarly, for the case g = Sn, we write (L∼, R∼,C∼0 ,C∼, δ∼, d∼,6∼,M∼) anddenote [a1, . . . , an] withouts the subscript for a g-tuple; we call such a tuplea symmetric tuple.

3.1.2 Isomorphism of the pregeometries of M 6∼ and Mg

Let n ∈ N be fixed. We will use Proposition 2.5.3 in order to show that thepregeometries associated with (C6∼, δ6∼) and (Cg, δg) are isomorphic, for anyg ≤ Sn. Fix some g ≤ Sn until the end of this section.

(Cg, δg) geometrically extends to (C 6∼, δ6∼)

Definition 3.1.8. Let X be some set and let R ⊆ Xn. We say a setdesym(R) ⊆ Xn is a g-desymmetrization of R if (desym(R) ∩ [x1, . . . , xn]g)is of size exactly 1 for every (x1, . . . , xn) ∈ R.

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When writing desym(R), we implicitly mean some arbitrary g-desymmatrizationof R.

Observation 3.1.9. If A ∈ Cg then | desym(Rg(A))| = rg(A).

Definition 3.1.10. Let B ∈ Cg and let A ⊆ B. Let A ∈ C 6∼ be a structurewith universe A. We construct B 6∼A ∈ C

6∼0 , a superstructure of A.

B 6∼A := B

R 6∼(B 6∼A) := R6∼(A) ∪ desym(RBg (B/A))

We call B 6∼A a desymmetrization of B over A.

Informally, a desymmetrization of B over A emulates the way B is relatedto AB, but does so over A, disregarding the structure induced on the set Aby B. Any g-relations are then replaced by ordinary tuples, so the resultingstructure is in C6∼0 .

Lemma 3.1.11. Let B ∈ Cg be a structure with universe B. Let A ⊆ B andlet A ∈ C6∼ be a structure with universe A. Let B 6∼A be a desymmetrization of

B over A. Then δBg (X/X ∩ A) = δB 6∼A6∼ (X/X ∩ A) for all X ⊆ B.

If in addition A 6g B, then A 6 6∼ B 6∼A and B 6∼A ∈ C 6∼.

Proof. The first part is immediate from the construction and Observation3.1.9. The additional part follows by our assumption.

Recall Definition 2.5.1 for the next result.

Proposition 3.1.12. (Cg, δg) (C6∼, δ6∼).

Proof. Let A1 ∈ Cg, A2 ∈ C6∼ be structures with universe A such thatdA1g (X) = dA2

6∼ (X) for all X ⊆ A. Let B ∈ Cg with A1 6g B.

Denote D = B 6∼A2some desymmetrization of B over A. By the additional

part of the above lemma, D ∈ C6∼ and A2 6 6∼ D. By the main statement ofthe above lemma, for any Y ⊆ B the equality δBg (Y/Y ∩ A) = δD6∼(Y/Y ∩ A)holds. By Lemma 2.5.4.iii, dD6∼(X) = dBg (X) for all X ⊆ B, and so B and Dalong with the identity map are the structures sought for.

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(C 6∼, δ6∼) geometrically extends to (Cg, δg)

Imitating the pregeometry of a structure A ∈ C6∼ with a structure from Cg isless obvious. The difficulty is that the predimension of a structure of size nmay be as low as zero in C6∼, but if g is of index strictly smaller than n, thisis not the case in Cg. We resolve this difficulty by attaching to each relatedtuple (a1, . . . , an) a new element e(a1,...,an), allowing us more freedom withoutchanging the pregeometry of the original structure.

Notation. For the remainder of this section, we use a and (a1, . . . , an) in-terchangeably.

Definition. Let B ∈ C6∼ and let A ⊆ B. Define:

• newel(B/A) = {e(a) | a ∈ RB6∼(B/A)}

• newrel(B/A) = {(e(a), a1, . . . , an−1) | e(a) ∈ newel(B/A)}

Definition 3.1.13. Let B ∈ C6∼ and let A ⊆ B. We define BrlxA , a super-structure of B:

BrlxA = B ∪ newel(B/A)

R6∼(BrlxA ) = R6∼(B) ∪ newrel(B/A).

Observation. Let B ∈ C 6∼ and let A ⊆ B. Then B 6 6∼ BrlxA and conse-quently BrlxA ∈ C6∼

Definition. Let B ∈ C6∼ and let A ⊆ B. Define the set

Adjust(B/A) = {[e(a), a1, . . . , an−1]g, [e(a), a2, . . . , an]g : e(a) ∈ newel(B/A)}

Definition 3.1.14. Let B ∈ C6∼ and let A ⊆ B. Let A ∈ Cg be a structurewith universe A. We construct Bg

A ∈ Cg0, a superstructure of A:

BgA = B ∪ newel(B/A)

Rg[BgA] = Rg[A] ∪ Adjust(B/A).

Lemma 3.1.15. Let C ∈ C6∼ and let A ⊆ C. Let A ∈ Cg be a structure withuniverse A. Denote B = CrlxA and D = CgA. Denote the common universe ofB and D by B. Then:

• If Y ⊆ B is d 6∼-closed in B then δB6∼(Y/Y ∩ A) = δDg (Y/Y ∩ A).

37

• If Y ⊆ B is dg-closed in D then δB6∼(Y/Y ∩ A) = δDg (Y/Y ∩ A).

• If A 66∼ C then A 6g D and D ∈ Cg.

Proof. We say that Y ⊆ B is good, if for every e(a) ∈ newel(C/A), if Y ∩{a1, . . . , an} is of size at least n − 1, then a1, . . . , an, e

(a) ∈ Y . Note that ifY ⊆ B is either d6∼-closed in B or dg-closed in D, then it is good.

Let Y ⊆ B be good. In order to prove the first two points, it is enoughto prove that rB6∼(Y/Y ∩A) = rDg (Y/Y ∩A). Observe that by construction ofB and the assumption on Y :

rB6∼(Y/Y ∩ A) =∑

a∈RC6∼(Y ∩C/A)

|{a, (e(a), a1, . . . , an−1)}| = 2 · rC6∼(Y ∩ C/A)

Again, by construction of D and the assumption on Y , clearly:

rDg (Y/Y ∩ A) =∑

a∈RC6∼(Y ∩C/A)

rDg (ae(a)) = 2 · rC6∼(Y ∩ C/A)

We prove the third point. Assume now that A 6 6∼ C. Since C 66∼ B bythe above observation, by transitivity we have A 6 6∼ B. Let A ⊆ Y0 ⊆ Band define for every i ∈ N

Yi+1 = Yi ∪ {a1, . . . , an, e(a) : a ∈ RC6∼(C/A), |Yi ∩ {a1, . . . , an}| ≥ n− 1}

Denote Y =⋃i∈N Yi. By construction of D, δDg (Y ) ≤ δDg (Y0). This implies

that δDg (Y/A) ≤ δDg (Y0/A). As Y is good, by A 6 6∼ B and what we showed,0 ≤ δB6∼(Y/A) = δDg (Y/A). In conclusion, δDg (Y0/A) ≥ 0. Then A 6g D.

Proposition 3.1.16. (C6∼, δ6∼) (Cg, δg)

Proof. Let A1 ∈ C6∼, A2 ∈ Cg be structures with universe A such thatdA16∼ (X) = dA2

g (X) for all X ⊆ A. Let C ∈ C6∼ with A1 6 6∼ C.Denote B = CrlxA and D = CgA2

. Denote the common universe of B andD by B. By the above lemma, D ∈ Cg and A2 6g D. By the first twoitems of the above lemma and Lemma 2.5.4.iii, we have dB6∼(X) = dD∼(X) forall X ⊆ B. Thus, B and D along with the identity map are the structuressought for.

Theorem 3.1.17. For every g ≤ Sn, PG(Cg, δg) ∼= PG(C6∼, δ6∼).

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3.2 Reduction relations between the struc-

tures Mg

For this section we let n ∈ N≥3 be fixed1. We will prove that for any h, g ≤ Sn,the structure Mg is isomorphic to a proper reduct of Mh.

Remark. We will show thatMh andMg are reducts of one another. However,the reduct we find is proper, hence the more common model-theoretic ques-tion of whether Mg and Mh are bi-interpretable stands. Bi-interpretability(in the sense of reducts, rather than arbitrary interpretations) here wouldmean that Mh is isomorphic to an improper reduct of Mg, hence Rg is adefinable relation in Mh and Rh is a definable relation in Mg.

3.2.1 Reduction of Mh to Mg for h ≤ g

Fix some subgroups h, g ≤ Sn such that h ≤ g. Consider the Lh-formula

ϕRg(x1, . . . , xn) :=∨σ∈g

Rh(xσ(1), . . . , xσ(n))

For an Lh-structure M with age(M) ⊆ Ch, let M+ be the definable expan-sion of M to Lh ∪ Lg where

M+ |= ∀x1, . . . , xn(Rg(x1, . . . , xn)↔ ϕRg(x1, . . . , xn))

and denote MRg =M+ � Lg. Note that age(M) ⊆ Cg0 by definition of ϕRg .

We will show that Mg∼= (Mh)Rg and that h � g implies (Mh)Rg is a proper

reduct. For the remainder of this subsection we use the definitions and resultsof Section 2.6 with respect to our choice of ϕRg .

Observation 3.2.1. If A ⊆M for some structure with age(M) ⊆ Ch, thenAMRg = ARg . In particular, 6h encloses reducts. (Definition 2.6.1)

Lemma 3.2.2. For any A ∈ Ch, δh(A) ≤ δg(ARg).

1Some proofs in this section do not work for n ≤ 2, hence the requirement that n ≥ 3.The underlying reason for this, is that for n ≤ 2 the pregeometry associated to the resultingstructures is disintegrated.

39

Proof. Explicitly, ARg is the structure with universe A and

Rg[ARg ] = {[x1, . . . , xn]g : [x1, . . . , xn]h ∈ Rh[A]}Thus by definition, rh(A) ≥ rg(ARg) and the lemma is evident.

Corollary 3.2.3. Ch reduces to Cg (Definition 2.6.2)

Proof. Let A ∈ Ch. We have already noted that ARg ∈ Cg0, and by the above

lemma, clearly ARg ∈ Cg.

Lemma 3.2.4. (Ch, δh) has a mixed amalgam for (Cg, δg). (Definition 2.6.6)

Proof. Let A ∈ Ch and let B ∈ Cg with ARg 6g B. Choose some sectionS ⊆

⋃Rg[B/ARg ] such that |S ∩ [a1, . . . , an]g| = 1 for each [a1, . . . , an]g ∈

Rg[B/ARg ]. Let C be the structure:

C := B

Rh[C] := Rh[A] ∪ {[a1, . . . , an]h | (a1, . . . , an) ∈ S}Just as in Lemma 3.1.11 and the short discussion preceding it, becauseARg 6g B, we have A 6h C and C ∈ Ch. By construction, it is clear thatCRg = B and in particular B 6∼ CRg .

Lemma 3.2.5. If h � g, then (Ch, δh) is lenient towards (Cg, δg). (Definition2.6.8)

Proof. Let F ∈ Ch. Let a1, . . . , an /∈ F be distinct new elements. Letσ ∈ g \ h.Define the structure A with

A = F ∪ {a1, . . . , an}Rh[A] = Rh[F ] ∪ [a1, . . . , an]h

Define the structure B with

B = F ∪ {a1, . . . , an}Rh[B] = Rh[F ] ∪ {[a1, . . . , an]h, [aσ(1), . . . , aσ(n)]h}

Clearly A and B fulfil the requirements in the definition.

Theorem 3.2.6. If h ≤ g, then the structure Mg is isomorphic to a reductof Mh. If in addition h � g, then Mg is isomorphic to a proper reduct of Mh.

Proof. All conditions of Proposition 2.6.7 hold with respect to (Ch, δh) and(Cg, δg). Thus, (Mh)Rg

∼= Mg.If additionally h � g, then the conditions of Lemma 2.6.9 also hold and

(Mh)Rg is a proper reduct of Mh.

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3.2.2 Exquisite formulas

In this subsection and the next, we construct a formula which allows us toreduce from M∼ to M 6∼.

Notation 3.2.7. For a tuple a = (a1, . . . , am) we write s(a) for the set ofelements appearing in a. i.e. s(a) = {a1, . . . , am}.

Recall that n is the arity of R∼. We limit our discussion to the class ofL∼-structures whose age is a subclass of C∼0 .

Definition 3.2.8. Let q(x; y) be a complete L∼ atomic type of xy. For atuple ab with |ab| = xy we define

Gq(a; b) = {[c1, . . . , cn] : R∼(c1, . . . , cn) ∈ q(a; b)}

We denote dq = | s(ab)| − |Gq(a; b)|.

Definition 3.2.9. Let q(x; y) be a complete L∼ atomic type of xy

i. We say that q is nice if q(x; y) implies that the elements of xy are pairwisedistinct, |x| = n, |y| ≥ 2n, ¬R∼(x) ∈ q(x; y) and dq = n− 1.

ii. We say that q is intertwined if δ∼(s(ab)/X) < 0 whenever q(a; b) holds,X ⊂ s(ab), and |X| > n.

iii. We say that q is without symmetry if q(x; y) = q(u; v)2 implies xy = uv.

iv. We say that q is exquisite if it is nice, intertwined and without symmetry.

Observation 3.2.10. For q(x; y) nice, |Gq(x; y)| = |y|+ 1 > 2n

Notation 3.2.11. For q(x; y) a complete L∼ atomic type of xy, denote byq+(x; y) the formula

q+(x; y) :=∧

R∼(z1,...,zn)∈q(x;y)

R∼(z1, . . . , zn)

where the zi in the definition are elements of xy.

Remark. The atomic formulas in the conjuction q+(x; y) correspond to therelations in Gq(a; b).

2Equality here is as sets of formulas

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Observation 3.2.12. If p(x, y) is a complete L∼ atomic type implyingq+(x; y) for some q(x; y) intertwined, then p(x, y) is intertwined.

Notation 3.2.13. For q(x; y), a complete atomic type of xy, denote byψq(x; y) the formula stating that for every uv such that q+(u; v) holds, ifxy 6= uv, then | s(xy) ∩ s(uv)| ≤ n.

For q an atomic type, we denote q(x; y) = q(x; y) ∧ ψq(x; y). When wesay that q is an exquisite formula, we mean that q is exquisite.

Fix q(x; y), an exquisite atomic type, until the end of this subsection.

Definition 3.2.14.

i. For R a set of symmetric n-tuples we say that ab is adjacent to R ifq(a; b) holds and Gq(a; b) ∩R 6= ∅.

ii. We say that ab and cd distinct are adjacent if q(a; b) and q(c; d) hold,and Gq(a; b) ∩Gq(c; d) 6= ∅.

iii. We say that q is asocial if |Gq(a; b) ∩ Gq(c; d)| = 1 whenever ab, cd areadjacent.

Observation 3.2.15. If q(a; b) and q(c; d) hold, then by ψq(a; b) and q+(c; d)holding we have that there are at most n elements in s(ab) ∩ s(cd). Hence,in Gq(a; b) ∩Gq(c; d) there is at most one n-tuple, containing all elements ofs(ab) ∩ s(cd). Conclude that an exquisite formula q(x; y) is asocial.

Definition 3.2.16. For R, a set of symmetric n-tuples, call a sequence ofdistinct tuples (a1b1, . . . , akbk) an R-adjacency-chain if

• a1b1 is adjacent to R

• aibi is adjacent to ai+1bi+1 for 1 ≤ i < k.

Definition 3.2.17. For R, a set of symmetric n-tuples, call an R-adjacency-chain (a1b1, . . . , akbk) an R-adjacency-loop if

• If k = 1, then |R ∩Gq(ak; bk)| ≥ 2

• If k > 1, denoting the unique symmetric tuple in Gq(ak−1; bk−1) ∩Gq(ak; bk) by r, we have that akbk is adjacent to (R∪

⋃k−2i=1 G

q(ai; bi)) \{r}

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We say that (a1b1, . . . , akbk) is proper if in addition Gq(ak; bk) * R

Lemma 3.2.18. LetM be an L∼-structure with age(M) ⊆ C∼0 . Let B ⊆Mbe finite, R ⊆ RM∼ [B] and let (a1b1, . . . , akbk) be some R-adjacency-chain,then

i. |B ∪⋃ki=1 s(aibi)| − |R ∪

⋃ki=1G

q(ai; bi)| ≤ |B| − |R|

ii. If (a1b1, . . . , akbk) is a proper R-adjacency-loop, the above inequality isstrict.

Proof.i. We prove by induction on k:k = 1: If s(a1b1) ⊆ B then the statement is clear, assume this is not the

case. We have R ∩ Gq(a1; b1) 6= ∅, which implies |B ∩ s(a1b1)| ≥ n. Let x ∈s(a1b1)\B, then because q is intertwined, s(a1b1) is in the self-sufficient closureof (Bx ∩ s(a1b1)) and, moreover, | s(a1b1) \ Bx| − |Gq(a1; b1) \ R| < 0. Since| s(a1b1)\B| = | s(a1b1)\Bx|+1, this gives us | s(a1b1)\B|−|Gq(a1; b1)\R| ≤ 0and thus the statement.

k > 1: Denote B1 = B∪s(a1b1) and R1 = R∪Gq(a1; b1). By the previouscase, since (a1b1) is anR-adjacency-chain, we have that |B1|−|R1| ≤ |B|−|R|.Now, (a2b2, . . . , akbk) is an R1-adjacency-chain of length k−1, so by inductionhypothesis |B1 ∪

⋃ki=2 s(aibi)| − |R1 ∪

⋃ki=2G

q(ai; bi)| ≤ |B1| − |R1|. Thestatement immediately follows.

ii. Assume first that |R ∩Gq(ak; bk)| ≥ 2. Then | s(akbk) ∩B| > n. Sinceq is intertwined and Gq(ak; bk) * R, we have | s(akbk) \ B| − |Gq(ak; bk) \R| < 0. Denoting Bk = B ∪ s(akbk) and Rk = R ∪ Gq(ak; bk), we have|Bk| − |Rk| < |B| − |R|. If k = 1, then we are done. Otherwise, notethat (a1b1, . . . , ak−1bk−1) is an Rk-adjacency-chain, so by what we proved|Bk ∪

⋃k−1i=1 s(aibi)| − |Rk ∪

⋃k−1i=1 G

q(ai; bi)| ≤ |Bk| − |Rk|, and we are done.We can now prove the general statement inductively.If k = 1, then |R ∩ Gq(ak; bk)| ≥ 2 and we just proved the statement for

this case. So we may assume k > 1 and |R ∩Gq(ak; bk)| < 2.Denote B1 = B ∪ s(a1b1) and R1 = R ∪ Gq(a1; b1), then as before

|B1| − |R1| ≤ |B| − |R|. It is clear that (a2b2, . . . , akbk) is an R1-adjacency-loop. Since q is asocial, |Gq(a1; b1) ∩ Gq(ak; bk)| ≤ 1. By definition of R1,we thus have |R1 ∩ Gq(ak; bk)| ≤ 2. Since |Gq(ak; bk)| ≥ 3, this meansGq(ak; bk) * R1 and so (a2b2, . . . , akbk) is proper. By induction hypothe-

sis, |B1 ∪⋃ki=2 s(aibi)| − |R1 ∪

⋃ki=2 G

q(ai; bi)| < |B1| − |R1| and the proof iscomplete.

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Definition 3.2.19. Let M be some L∼-structure with age(M) ⊆ C∼0 .Say that (ab, cd) is a collision in M if ab is adjacent to cd. Define cM to

be the number of collisions in M.Say that (ab, cd) is a weak-collision in M if M |= q+(a; b) ∧ q+(c; d) and

Gq(a; b) ∩ Gq(c; d) 6= ∅. Define wM to be the number of weak-collisions inM.

Definition 3.2.20. Let M be some L∼-structure with age(M) ⊆ C∼0 . Wesay that r ∈ R∼[M] is q-unique in M if

• There exists a unique tuple arbr ∈ M such that M |= q(ar; br) andr ∈ Gq(ar; br).

• The tuple arbr appears in a collision in M.

For N an L∼-structure with N ⊆ M and R∼[N ] ⊆ R∼[M], we say thatr ∈ R∼[N ] is q[M]-unique in N if

• There exists a unique tuple arbr ∈ N such that M |= q(ar; br), r ∈Gq(ar; br) and Gq(ar; br) ⊆ R∼[N ].

• The tuple arbr appears in a collision in M.

Lemma 3.2.21. Let A ∈ C∼ be such that cA > 0, then there is some r ∈R∼[A] which is q-unique in A.

Proof. Assume that the statement is false. Let A ∈ C∼ be such that cA > 0and there is no q-unique tuple in A.

Claim. Consider any L∼-structure B with universe B ⊆ A and R∼[B] =RB ⊆ RA∼[B]. If there is some ab ∈ B with Gq(a; b) ⊆ RB and some r1 ∈Gq(a; b) q[A]-unique in B, then there is in A a proper RB-adjacency-loop(a1b1, . . . , akbk) with r1 ∈ Gq(a1; b1) and a1b1 6= ab.

Proof. Since r1 is not q-unique in A, there is in A some a1b1 adjacent to absuch that r1 ∈ Gq(a1; b1). Let r2 ∈ Gq(a1; b1)\{r1}. Since r2 is not q-unique inA, there is in A some a2b2 adjacent to a1b1 with r2 ∈ Gq(a2; b2). Continuingin this manner, choosing ri ∈ Gq(ai; bi) \ {ri−1}, for every k we have that(a1b1, . . . , akbk) is an RB-adjacency-chain. Since A is finite, there is somelarge enough l such that albl = ambm for some m < l. So by construction,(a1b1, . . . , al−1bl−1) is an RB-adjacency-loop.

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Now let k be minimal such that (a1b1, . . . , akbk) is an RB-adjacency-loop.Since r1 is q[A]-unique in B, it must be that Gq(a1; b1) * RB. If, howeverGq(ak; bk) ⊆ RB, then (a1b1, . . . , ak−1bk−1) is an RB-adjacency-loop in con-tradiction to the minimality of k. So Gq(ak; bk) * RB and so (a1b1, . . . , akbk)is proper.

Now, fix some ab ∈ A such that ab appears in a collision in A. Denotetq = |Gq(a; b)|. We describe an inductive process. Denote X0 = s(ab),R0 = Gq(a; b) = {r1, . . . , rtq} and let A0 be the L∼-structure with universeX0 and R∼[A0] = R0. Note that all of {r1, . . . , rtq} are q[A]-unique in A0

and tq > 2n. In particular, at least (2n + 1) of {r1, . . . , rtq} are q[A]-uniquein A0.

Let i < n and assume that Xi ⊆ A and R0 ⊆ Ri ⊆ RA∼[Xi] are givenand that, in the structure Ai with universe Xi and R∼[Ai] = Ri, at least(2(n− i) + 1) elements of {r1, . . . , rtq} are q[A]-unique in Ai.

By the claim, choose some proper Ri-adjacency-loop (a1b1, . . . , akbk) withk minimal, such that a1b1 6= ab, and rj ∈ Gq(a1; b1) for some 1 ≤ j ≤ tqsuch that rj is q[A]-unique in Ai. Define Xi+1 = Xi ∪

⋃ki=1 s(aibi), Ri+1 =

Ri∪⋃ki=1G

q(ai; bi) andAi+1 the structure with universe Xi+1 and R∼[Ai+1] =Ri+1.

In order to proceed with the inductive construction, we only need to showthat at least (2(n − i − 1) + 1) elements of {r1, . . . , rtq} are q[A]-unique inAi+1. We show that if m 6= j is such that rm is q[A]-unique in Ai but not inAi+1, then rm ∈ Gq(ak; bk).

Let cd be such that rm ∈ Gq(c; d) and Gq(c; d) ⊆ Ri+1, but Gq(c; d) * Ri.Assume cd 6= akbk. By minimality of k, also cd 6= aibi for every i < k.If Gq(c; d) ∩ Ri = {rm}, then because Gq(c; d) ⊆ Ri+1, by asociality of q,it must be that cd is adjacent to tq − 1 many tuples of (a1b1, . . . , akbk) incontradiction to the minimality of k. So (cd) is a proper Ri-adjacency-loop.

Let l be minimal such that (cd) is a proper Rl-adjacency-loop and notethat l ≤ i and |Gq(c; d) ∩ Rl| = 2. Since at each stage of the constructionwe choose a proper adjacency-loop of minimal length, we have that for everyl ≤ s ≤ i, in the s-th stage we have chosen a proper adjacency-loop of length1, say (csds). Therefore, by asociality of q, if cd 6= csds, then |Gq(c; d)∩(Rs+1\Rs)| ≤ 1. We know that Gq(c; d) * Ri, thus cd 6= csds for all s < i, andconsequently |Gq(c; d)∩Ri| ≤ (i−l)+2 ≤ (n−1)+2 < tq−1. Now, noting thatcidi = akbk, as |Gq(c; d)∩Ri+1| = tq, it must be that |Gq(c; d)∩Gq(ak; bk)| ≥ 2.By asociality, this is a contradiction and so cd = akbk, and rm ∈ Gq(ak; bk).

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Now, by asociality of q, we have |Gq(ak; bk) ∩ Gq(a; b)| ≤ 1. Thus, mis uniquely determined and any symmetric tuple r ∈ {r1, . . . , rtq} \ {rj, rm}which was q[A]-unique in Ai is also q[A]-unique in Ai+1, in particular, atleast (2(n− i− 1) + 1) elements.

So we are able to construct this way A0, . . . ,An. By construction andLemma 3.2.18, we have |Xi+1|−|Ri+1| < |Xi|−|Ri| which is exactly δ∼(Ai+1) <δ∼(Ai). Inductively this gives us δ∼(An) ≤ δ∼(A0)−n. It is only left to recallthat δ∼(A0) = dq < n and then

δA∼(Xn) ≤ δ∼(An) ≤ δ∼(A0)− n < 0

So A /∈ C∼ in contradiction to its choice. We conclude that there must besome q-unique symmetric tuple in any A ∈ C∼ with cA > 0.

Lemma 3.2.22. If D ∈ C∼0 is a simple amalgam3 of B1, . . . ,Bk over A with|A| ≤ n and ab ∈ D is such that D |= q+(a; b), then s(ab) ⊆ Bi for some1 ≤ i ≤ k.

Proof. Assume that the statement is false and let D with ab ∈ D be a counterexample. Denote X = s(ab). We may assume that RD∼[X] = Gq(a; b), sinceD remains a counter example to the statement even after removing the edgesRD∼[X] \Gq(a; b) from the structure.

For every i ≤ k, the structure D can be seen as the simple amalgam (overA) of Bi and the simple amalgam of B1, . . . ,Bi−1,Bi+1, . . .Bk over A. Thus,we may assume that k = 2, by choosing i such that X ∩Bi * A.

Denote Xi = X ∩ Bi. Since |X ∩ A| ≤ n and |X| > 2n, without lossof generality |X1| > n. Since q is intertwined and D is a simple amalgam,δ∼(X2/X ∩ A) = δD∼(X2/X1) < 0. So δD∼(X2) < δD∼(X ∩ A) ≤ |X ∩ A|.

Choose arbitrarily some Y ⊆ X1 \ A with |Y | = (n − |X ∩ A|), thenδD∼(X2Y ) < |X ∩ A| + |Y | = n. As |X2Y | > n and q is intertwined,δD∼(X/X2Y ) < 0. Thus, δD∼(X) ≤ n − 2 in contradiction to δD∼(X) = dq =n− 1.

Proposition 3.2.23. Let A ∈ C∼ with cA > 0. Then there exists someB ∈ C∼ with A ⊆ B, wB < wA and such that if A |= ∃yq(a; y), thenB |= ∃yq(a; y), for every a ∈ A.

3in the case of C∼0 , a simple amalgam is a free join

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Proof. LetA ∈ C∼ with cA > 0. By Lemma 3.2.21 there exists some q-uniquer ∈ R∼[A]. Let ab ∈ A be the unique tuple such that r ∈ Gq(a; b). Let w bea tuple of new elements with |b| = |w|. We define a new L∼-structure B asfollows:

B = A ∪ s(w)

R∼[B] = (R∼[A] \ {r}) ∪Gq(a; w)

Note that B is a simple amalgam of AB and s(aw)B over s(a)B.

Claim. B ∈ C∼.

Proof. Assume the contrary, then there is some X ⊆ B such that δB∼(X) < 0.Choose X to be such that δB∼(X) is minimal. If X ⊆ A, then by constructionδB∼(X) ≥ δA∼(X) ≥ 0. So it must be X * A. Let c ∈ X ∩ s(w), since δB∼(X)is minimal, there is some symmetric tuple in RB∼[X] in which c appears. Inparticular, by construction, |X∩s(aw)| ≥ n, and so δB∼(s(aw)/X) ≤ 0. Thus,we may assume s(aw) ⊆ X.

Because δB∼(s(aw)) = n−1 and δB∼(X) < 0, we have δB∼(X/ s(aw)) < 1−n.As B is a simple amalgam, δB∼(X∩A/ s(a)) < 1−n and therefore δA∼(X∩A) ≤δB∼(X∩A) ≤ 0. If s(b) ⊆ X then by construction δA∼(X∩A) < δB∼(X∩A) ≤ 0which is impossible, so s(b) * X. But s(b) is in the self-sufficient closure ofs(a) in A, so X ∩ A is not self-sufficient in A, which is a contradiction toδA∼(X∩A) ≤ 0. We conclude that δB∼(X) ≥ 0 for any X ⊆ B and B ∈ C∼.

Observe that for every cd such that B |= q+(c; d), by Lemma 3.2.22 eithers(cd) ⊆ s(aw) or s(cd) ⊆ A. If s(cd) ⊆ s(aw), then since q is withoutsymmetry, cd = aw. By construction of B, if s(cd) ⊆ A, then alreadyA |= q+(c; d).

Claim. wB < wA.

Proof. Let (c1d1, c2d2) be a weak-collision in B. If s(cidi) ⊆ A for i ∈ {1, 2},then by what we showed this is already a weak collision in A. Otherwise, itmust be that one of the tuples in the weak-collision is aw and so s(c1d1) ∩s(c2d2) ⊆ s(a). This, however, cannot be, because B |= ¬R∼(a). So we havewB ≤ wA. Now, since r is q-unique in A, it was involved in a collision, whichno longer exists in B because B |= ¬q+(a; b). We conclude that wB < wA.

It remains to show that for c ∈ A, if A |= ∃yq(c; y), then B |= ∃yq(c; y).

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For a, we have that B |= q(a; w) and we’ve seen that if B |= q+(u; v) foruv 6= aw, then | s(uv) ∩ s(aw)| ≤ n, so B |= q(a; w) by definition.

Let cd ∈ A with c 6= a and such that A |= q(c; d). Assume that B |=¬q(c; d). Since r was q-unique, it is still the case that B |= q(c; d). So itmust be that B |= q+(u; v) for some uv ∈ A with | s(cd) ∩ s(uv)| > n. Butwe’ve seen that in that case, also A |= q+(u; v) which implies A |= ¬q(c; d)in contradiction. So it must be B |= q(c; d). This concludes the proof.

3.2.3 Constructing an exquisite formula

In this subsection, we prove the existence of an exquisite formula which issuitable for our needs. We do this by inducting on the arity of the relationR∼. Therefore, we denote by Lk∼ = {Rk

∼} the language of a symmetric k-aryrelation.

Lemma 3.2.24. If for k ∈ N≥3 there exists an exquisite Lk∼-formula, thenthere exists some exquisite Lk+1

∼ -formula.

Proof. Let q(x; y) be an exquisite Lk∼ formula. Denote a = (a1, . . . , ak),|y| = l, and b = (b1, . . . , bl). Let A be the Lk∼-structure with universe s(ab)such that A |= q(a; b). Fix some r ∈ Rk

∼[A], say r = [t1, . . . , tk]. Sincel > 2k, there are some j1, . . . , jk+1 distinct such that for each 1 ≤ i ≤ k + 1,the element bji does not appear in r. By reordering y, we may assume ji = i,hence, bi does not appear in r for 1 ≤ i ≤ k + 1.

Let ak+1, c1, . . . , ck+1 be new elements. For every i ∈ Z, denote ci =ci(mod k+1). Define

Γ1 = {[c1, x1, . . . , xk] : [x1, . . . , xk] ∈ Rk∼[A] \ {r}}

Γ2 = {[ak+1, bi, ci, . . . , ci+(k−2)] : 1 ≤ i ≤ k + 1}

Consider the Lk+1∼ -structure B defined as follows:

B = {a1, . . . , ak+1, b1, . . . , bl, c1, . . . , ck+1}Rk+1∼ [B] = Γ1 ∪ {[c2, t1, . . . , tk]} ∪ Γ2

Claim 1. The structure B has no non-trivial automorphisms.

Proof. The element c1 is definable by virtue of being the unique elementappearing in at least |Rk

∼[A]|+ 1 relations in Rk+1∼ [B].

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Now, ak+1 is definable as the unique element appearing in exactly k + 1distinct symmetric tuples in Rk+1

∼ [B], with exactly two of these tuples notcontaining c1. Hence, the set {c1, . . . , ck+1} is definable as the set of elementsappearing with ak+1 in at least two symmetric tuples in Rk+1

∼ [B].So s(ab) is definable as the complement of {ak+1, c1, . . . , ck+1}, and any au-

tomorphism of B induces a bijection on s(ab), which will be an automorphismof A. Since q(x; y) is without symmetry, A has no non-trivial automorphism.Thus, any automorphism of B fixes s(ab) pointwise.

Each element of the set {c2, . . . , ck+1} is definable over {b1, . . . , bk+1},since we can tell apart the relations in Γ2 using the bi’s. We’ve seen thatthe set {b1, . . . , bk+1} is fixed pointwise by any automorphism of B, and thusso is {c2, . . . , ck+1}. We conclude that the only automorphism of B is theidentity.

Claim 2. If X ⊂ B with |X| > k + 1, then δB∼(B/X) < 0.

Proof. Let X 6∼ B with |X| > k+1 be of minimal size. Showing that X = Bwill prove the claim. Denote RX = Rk+1

∼ [XB]. Note that δ∼(B) = dq +1 = k.This implies δB∼(X) ≤ k and therefore |RX | ≥ 2.

First we show that it cannot be that RX ⊆ Γ1. Assume the contrary anddenote X ′ = X \ {c1}. By minimality of X, it must be that X ′ ⊆ s(ab). IfX ′ = s(ab), then by construction δB∼(X) = dq + 2. If X ′ 6= s(ab), then since|X ′| > k and q(x; y) is intertwined, δA∼(X ′) > dq. By construction, we haveδB∼(X) > δA∼(X ′). Thus, in any case δB∼(X) > dq + 1 in contradiction to theself-sufficiency of X.

Now we show that |X| > k + 2. Assume |X| = k + 2 to the contrary.Let r1, r2 ∈ RX be distinct and arbitrary, then by the pigeon-hole principler1 and r2 have exactly k elements in common. By construction, this cannotoccur unless RX ⊆ Γ1 which we proved is impossible. So |X| > k + 2.

Now, observe that for any x ∈ X, the element x must appear in at leasttwo distinct symmetric tuples in RX , or else X \ {x} is self-sufficient in Bwith |X \ {x}| > k + 1, in contradiction to the minimality of X.

If [c2, t1, . . . , tk] /∈ RX , then because RX * Γ1, we have RX ∩ Γ2 6= ∅. If[c2, t1, . . . , tk] ∈ RX , then there must be some r2 ∈ RX \ {r1} in which c2

appears. In particular, r2 ∈ RX ∩ Γ2 and again RX ∩ Γ2 6= ∅.In any case, it must be that ak+1 ∈ X. So there must be at least two

symmetric tuples in RX in which ak+1 appears. By construction, this meansthere is at most a unique 1 ≤ j ≤ k + 1 such that cj /∈ X. Assume such

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a j exists. If 1 < j < k + 1, then cj+1 ∈ X and so there are at least twosymmetric tuples in RX in which cj+1 appears. However, there is only onesymmetric tuple in B in which cj+1 appears and cj does not. So it must bethat cj appears in some tuple in RX and, therefore, cj ∈ X. If j = 1 orj = k + 1, then the same argument applied to cj−1 shows that cj ∈ X. Weconclude that {ak+1, c1, . . . , ck+1} ⊆ X.

Since c1, c2 ∈ X, by construction, δB∼(s(ab)/X) ≤ δA∼(s(ab)/X ∩ s(ab)).If RX ⊆ Γ2, then a simple check shows that δB∼(X) ≥ k + 2. Thus, theremust be at least two distinct symmetric tuples in RX \Γ2. This implies that|X ∩ s(ab)| > k. Since q(x; y) is intertwined, if s(ab) * X, then δA∼(s(ab)/X ∩s(ab)) < 0. But this cannot be, since by assumption X is self-sufficient in Band therefore 0 ≤ δB∼(s(ab)/X). We conclude that B ⊆ X.

Denote c = (c1, . . . , ck+1). Let p(x, xk+1, y, z1, . . . , zk+1) be the completeatomic type of (a, ak+1, b, c) in B. Then p is nice by construction, withoutsymmetry by Claim 1, and intertwined by Claim 2. Thus, p is an exquisiteLk+1∼ formula.

Lemma 3.2.25. There exists an exquisite L3∼ formula.

Proof. Let A = {a1, a2, a3, b1, . . . , b8}. Define R as follows:

R = {[a1, b1, b2], [a2, b2, b3], [a3, b1, b7],

[a1, b3, b4], [a2, b4, b5], [a3, b8, b3],

[a1, b5, b6], [a2, b6, b7],

[a1, b7, b8] }

Consider the L3∼ structure A with universe A and R3

∼[A] = R. Note thatδ∼(A) = 2

Claim 1. The structure A has no non-trivial automorphisms.

Proof. The element a1 is definable as it is the only element present in fourdistinct tuples in R. Then, a2 and a3 are definable for not appearing with a1

in the same tuple in R, and appearing in exactly two and three tuples in R,respectively.

The set {b1, b8} is now definable by the property of not being in a tuplewith a2 in R. Thus, b7 is definable as the unique element sharing a tuplein R both with b1 and with b8. Clearly all other elements are definable overa1, a2, a3, b7. Since all the elements of A are definable, any automorphism ofA is trivial.

50

Claim 2. If X ⊂ A with |X| > 3, then δA∼(A/X) < 0.

Proof. Let X 6∼ A with |X| > 3 and such that X is of minimal size.Showing that X = A will prove the claim. Denote RX = R3

∼[XA]. A briefexamination shows that it must be that |X| > 4. Thus, any x ∈ X is presentin at least two tuples in RX , or else (X \ {x}) 6∼ A in contradiction to theminimality of X.

By this reasoning, we have

a3 /∈ X =⇒ b1, b8 /∈ X =⇒ b2, b7 /∈ X =⇒ a2 /∈ X =⇒ b4, b5, b6 /∈ X

so it must be that a3 ∈ X. As explained, both tuples in R involving a3

are in RX , and consequently b1, b3, b7, b8 ∈ X. Again, since b1 ∈ X it mustbe that [a1, b1, b2] ∈ RX implying a1, b2 ∈ X. Because b2 ∈ X, it must be[a2, b2, b3] ∈ RX and a2 ∈ X. We have so far A \ {b4, b5, b6} ⊆ X ⊆ A. Abrief calculation shows that if X 6= A, then δA∼(X) > 2. As δ∼(A) = 2 andX 6∼ A, this cannot be.

Denote a = (a1, a2, a3), b = (b1, . . . , b8). Let q(x; y) be the completeatomic type of (a, b) in A. By construction, Claim 1 and Claim 2, q(x; y) isan exquisite L3

∼ formula.

Corollary 3.2.26. Let k ∈ N≥3. There exists an exquisite Lk∼ formulaq(x; y).

Proof. The proof is by induction on k. For the base case k = 3, the previousLemma guarantees the existence of such a formula q(x; y). The inductionstep is then exactly Lemma 3.2.24.

3.2.4 Proper reduction of M∼ to M 6∼

Fix some exquisite L∼ formula q(x; y) with tq > 2n, and dq = n − 1, asguaranteed by Proposition 3.2.26.

For an L∼-structure M with age(M) ⊆ C∼, let M+ be the definableexpansion of M to L∼ ∪ L 6∼ where

M+ |= ∀x(R6∼(x)↔ ∃yq(x; y))

and denote M(q) =M+ � L 6∼.We will show that M 6∼ ∼= M∼(q) and that M∼(q) is a proper reduct of M∼. Forthe remainder of this section we use the definitions and results of Section 2.6with respect to our choice of q.

51

Lemma 3.2.27. The relation 6∼ encloses reducts.

Proof. Let A 6∼M where age(M) ⊆ C∼. Let a ∈ A be arbitrary.Assume b ∈ M is such that M |= q(a; b). Because δM∼ (s(b)/ s(a)) = −1,

there is some b ∈ s(b) in the self-sufficient closure of s(a) in M. Since q isintertwined we have that s(ab) is in the self-sufficient closure of s(a)b, andthus in the self-sufficient closure of s(a). Since s(a) ⊆ A, we get s(b) ⊆ A.Clearly, M |= ψq(a; b) implies A |= ψq(a; b), so A |= q(a; b).

Now assume b ∈ A such that A |= q(a; b). Clearly M |= q(a; b), so weonly need to show M |= ψq(a; b). Assume not, let cd ∈ M be such thatab 6= cd, M |= q(c; d) and | s(ab) ∩ s(cd)| > n. Since q is intertwined, s(cd)is in the self-sufficient closure of s(ab), and so in A. So A |= ¬ψq(a; b) incontradiction. Thus, M |= q(a; b).

We conclude that A(q) |= R6∼(a) if and only if M(q) |= R6∼(a).

For the next lemma, recall definitions 3.2.14 and 3.2.19.

Lemma 3.2.28. If A ∈ C∼ then A(q) ∈ C 6∼.

Proof. Assume that the statement is false. So there exists some A, an L∼-structure with A ∈ C∼ and A(q) /∈ C6∼. Choose A such that wA is minimal.We claim that cA = 0. If not, then by Lemma 3.2.23 there is some B ∈ C∼with wB < wA and R6∼(A(q)) ⊆ R6∼(B(q)). Clearly, since A(q) /∈ C6∼, alsoB(q) /∈ C6∼ in contradiction to the minimality condition on A.

LetX ⊆ A be such that δA(q)

6∼ (X) < 0. Denote T = {ab | a ∈ X, A |= q(a; b)}and let X = X ∪

⋃ab∈T s(b). Since there are no collisions in A, if ab and cd

are distinct with A |= q(a; b)∧ q(c; d), then Gq(a; b) and Gq(c; d) are disjoint.We thus calculate:

δA∼(X) = (|X|+ |X \X|)− rA∼(X)

≤ |X|+ |⋃ab∈T

s(b)| − |⋃ab∈T

Gq(a; b)|

≤ |X|+∑ab∈T

| s(b)| −∑ab∈T

|Gq(a; b)|

= |X| − |T |

≤ |X| − r6∼(A(q)) = δA(q)

6∼ (X) < 0

52

This contradicts our assumption that A ∈ C∼. Thus, the statement of thelemma holds.

Lemma 3.2.29. (C∼, δ∼) has a mixed amalgam for (C6∼, δ6∼).

Proof. LetA ∈ C∼ and let B ∈ C6∼ withA(q) 66∼ B. Denote T = R6∼(B/A(q)).For each a ∈ T let va be a tuple of distinct elements not appearing in B.Consider the L∼-structure C with

C = B ∪⋃a∈T

s(va)

R∼[C] = R∼[A] ∪⋃a∈T

Gq(a; va)

We first assert that A 6∼ C. Let X 6∼ C be arbitrary, denote XA =X ∩ A, XB = X ∩ B. We show δC∼(X) ≥ 0. Note that for any a ∈ T , if| s(ava) ∩ X| ≥ n, then δC∼(s(ava)/X) ≤ 0, so we may assume s(ava) ⊆ X.Denote TX = RB6∼(XB/XA) and calculate

δC∼(X/XA) = |X \XA| − rC∼(X/XA)

≥ (|XB \XA|+ |⋃a∈TX

s(va)|)− |⋃a∈TX

Gq(a; va)|

= |XB \XA| − |TX | = δB6∼(XB/XA) ≥ 0

Since δC∼(XA) = δA∼(XA) ≥ 0, we get δC∼(X) ≥ 0. So A 6∼ C and bytransitivity of 6∼, also C ∈ C∼. We now have left to show that B 6 6∼ C(q).

Let cd ∈ C be such that C |= q+(c; d). We observe that either c ∈ T andd = vc, or s(cd) ⊆ A. Assume that the latter is false. Then Gq(c; d) * R∼[A].By construction, this must mean that s(cd) intersects s(va) for some a ∈ T .As C is a simple amalgam of (C \ s(va))

C and s(ava)C over s(a)C, by Lemma

3.2.22, it must be that s(cd) ⊆ s(ava). Since q is without symmetry, cd = ava.

It is immediate from the above paragraph that RC(q)6∼ (C/A) ⊆ R6∼(B).

Additionally, we claim that if a ∈ RC(q)6∼ (A) and b is such that C |= q(a; b),

then already A |= q(a; b), meaning a ∈ R6∼(B). We have shown that s(b) ⊆ Amust occur, and then it is clear that A |= q(a; b). Our claim follows fromthe observation that, by construction, A |= ¬ψq(a; b) implies C |= ¬ψq(a; b).Thus, R6∼(C(q)) ⊆ R6∼(B).

53

Now, showing R6∼(B) ⊆ R6∼(C(q)) gives us B ⊆ C(q) and thereby B 6 6∼ C(q).If a ∈ T , it is clear by what we’ve shown that C |= q(a; va) and thereforea ∈ R6∼(C(q)). So let a ∈ R6∼(A(q)), and let b ∈ A be such that A |= q(a; b).By construction, C |= q(a; b). If cd ∈ C is such that | s(ab) ∩ s(cd)| > n andC |= q+(c; d), then as we’ve seen s(cd) ⊆ A, but by construction this wouldmean A |= q+(c; d), in contradiction to A |= ψq(a; b). So no such cd existand therefore C |= q(a; b). In particular, a ∈ R6∼(C(q)). So we have shownR6∼(B) ⊆ R6∼(C(q)).

We proved A 6∼ C and B 66∼ C(q), so C is a structure as required in thedefinition of a mixed amalgam.

Lemma 3.2.30. (C∼, δ∼) is lenient towards (C 6∼, δ6∼).

Proof. Let F ∈ C∼. Let a be an n-tuple of distinct elements not in F and letA = F ∪s(a). Define R1 = R∼[F ] and R2 = R∼[F ]∪{[a]}. Let Ai be the L∼-structure with universe A and R∼[Ai] = Ri. Clearly F 6∼ Ai for i ∈ {1, 2}and A1 is not isomorphic to A2 over F . Since Ai is a simple amalgamof F and s(a)Ai over the empty set, by Lemma 3.2.22, if Ai |= q+(c; d),then s(cd) ⊆ F . So R6∼(Ai(q)) = R6∼(F(q)) for i ∈ {1, 2} and thereforeA1(q) = A2(q). In particular, A1(q) is isomorphic to A2(q) over F . So A1 andA2 are as required by the definition of lenience.

Theorem 3.2.31. The structure M 6∼ is isomorphic to a proper reduct of M∼.

Proof. All conditions of Proposition 2.6.7 and Lemma 2.6.9 hold with respectto (C∼, δ∼) and (C6∼, δ6∼). Thus, M∼(q) is a proper reduct and M 6∼ ∼= M∼(q).

Corollary 3.2.32. For any g, h ≤ Sn, the structure Mg is isomorphic to aproper reduct of Mh.

Proof. By Theorem 3.2.6, Mg is isomorphic to a reduct of M∼. By Theorem3.2.31, M∼ is isomorphic to a proper reduct of M 6∼. By Theorem 3.2.6 again,M 6∼ is isomorphic to a reduct of Mh.

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Chapter 4

The clique construction and thegeometric construction

4.1 The clique construction

Fix n ∈ N \ {0, 1}. Let LS be the language of a single n-ary relation S.Throughout, the relation S is irreflexive [S(x1, . . . , xn) =⇒

∧i 6=j xi 6= xj]

and symmetric [S(x1, . . . , xn) =⇒∧σ∈Sn S(xσ(1), . . . , xσ(n))].

Definition 4.1.1. Let A be some finite LS-structure.We say that K ⊆ A with |K| ≥ n is a clique if [K]n ⊆ S(A). We say that Kis a maximal clique if there is no clique K ′ ⊆ A such that K ′ ⊃ K. DefineM(A) to be the set of maximal cliques of A.

For an LS-structure A, the set M(A) determines S(A) and vice versa.Explicitly,

S(A) =⋃

K∈M(A)

[K]n.

Observation 4.1.2. For an LS-structure A and a substructure B ⊆ A,

M(B) = {K ∩B : K ∈ M(A), |K ∩B| ≥ n}.

Definition 4.1.3. Define Cclq0 to be the class of finite LS-structures A such

that whenever K1, K2 ∈ M(A) are distinct, then |K1 ∩K2| < n.

Remark. The above definition implies that in particular no related tuple iscontained in two distinct maximal cliques.

55

Remark 4.1.4. The requirement that the intersection of any two cliques isof size less than n is not necessary. If this requirement is dropped, one mustintroduce some method of ‘predicting’, for each clique of size n, in how manymaximal-cliques it is contained. This can be achieved by considering eachmaximal clique as an imaginary element. For more details see Section 4.4.

Notation 4.1.5. For a finite set X define |X|∗ = max{0, |X| − (n− 1)}

Definition 4.1.6. For every finite LS-structure A define

s(A) =∑

K∈M(A)

|K|∗

andλ(A) = |A| − s(A)

Observation 4.1.7. For A ∈ Cclq0 , whenever B ⊆ A and K ∈ M(B), there

is a unique extension of K to a maximal clique of A. In particular,

s(B) =∑

K∈M(A)

|K ∩B|∗

Lemma 4.1.8. The function λ : Cclq0 → Z is submodular. That is, letting

D ∈ Cclq0 and letting A,B ⊆ D, we have

λD(A ∪B) + λD(A ∩B) ≤ λD(A) + λD(B).

Proof. For each K ∈ MD(A∪B) let KA, KB, KAB denote K ∩A,K ∩B,K ∩(A ∩B) respectively. Observe that for each K ∈ MD(A ∪B) we have

|K|∗ + |KAB|∗ ≥ |KA|∗ + |KB|∗

Thus, by Observation 4.1.7,

sD(A ∪B) + sD(A ∩B) =∑

K∈MD(A∪B)

|K|∗ +∑

K∈MD(A∪B)

|KAB|∗

≥∑

K∈MD(A∪B)

|KA|∗ +∑

K∈MD(A∪B)

|KB|∗

= sD(A) + sD(B)

proving the statement.

56

Corollary 4.1.9. The function λ is a predimension function on Cclq0 .

We associate to λ a scheme of dimension functions Λ and a notion ofself-sufficiency 6s as in Chapter 2

Definition 4.1.10. Let A1,A2 ∈ Cclq0 and let B be a common induced sub-

structure with universe A1∩A2. Define the standard amalgam of A1 and A2

over B to be the unique LS-structure D whose universe is A1 ∪A2 such thatM(D) = M ∪M ′ where

M = {K ∈ M(A1) ∪M(A2) : |K ∩B| < n}M ′ = {K1 ∪K2 : K1 ∈ M(A1), K2 ∈ M(A2), |K1 ∩K2| ≥ n}

Lemma 4.1.11. The standard amalgam is a simple amalgam for Cclq0 (with

respect to λ).

Proof. Let D be the standard amalgam of A1 and A2 over B. Like in theproof of Corollary 3.1.6, it suffices to show λ(D/A1) = λ(A2/B). This isclear by construction.

Definition 4.1.12. Cclq = {A ∈ Cclq0 | ∅ 6s A}.

By Lemma 2.3.4, (Cclq, λ) is a (simple) Fraısse-Hrushovski amalgamationclass. We denote the Fraısse-Hrushovski limit of (Cclq, λ) by Mclq and recallthat it has the following defining property:

(∗) Whenever A 6s B ∈ Cclq and A 6s Mclq, there exists a strong embed-ding f : B →Mclq fixing A pointwise.

4.2 Mclq is a proper reduct of M 6∼

Definition 4.2.1. Define ϕ′S to be the L 6∼-formula

ϕ′S(x1, . . . , xn) :=∧i 6=j

xi 6= xj ∧ ∃!y1, . . . , yn−1

∧i≤n

R 6∼(y1, . . . , yn−1, xi)

and define ϕS(x1, . . . , xn) to be the Lns-formula stating that ϕ′S(x1, . . . , xn)holds, and if there exists some xn+1 such that ϕ′S(x1, . . . , xi−1, xi+1, . . . , xn+1)

57

holds for all i ≤ n, then there exist y1, . . . , yn such that R6∼(y1, . . . , yn−1, xi)holds for all i ≤ n. Explicitly

ϕS(x1, . . . , xn) := ϕ′S(x1, . . . , xn)∧(∀xn+1

n∧i=1

ϕS(x1, . . . , xi−1, xi+1, . . . , xn+1)⇒ ∃!y1, . . . , yn−1

n+1∧i=1

R6∼(y1, . . . , yn−1, xi)

)

Remark. The convoluted definition of ϕS is in order to avoid a specificpathology. Consider the structure A ∈ C 6∼ defined

A = {ai : 1 ≤ i ≤ n} ∪n⋃i=1

{bi1, . . . , bin−1}

R6∼(A) = {(bi1, . . . , bi3, aj | 1 ≤ i, j ≤ n, i 6= j}.

In 〈A,ϕ′S(A)〉, the set {a1, . . . , an} is a maximal clique. However, we want thecliques in the reduct to be only of the form R6∼(y1, . . . , yn−1,A). Replacingϕ′S with ϕS guarantees this will be the case.

Theorem 4.2.2. 〈M 6∼, ϕS(M 6∼)〉 ∼= 〈Mclq, S〉

Proof. By Lemma 2.6.7 it suffices to show:

1. For every A ∈ C6∼, we have 〈A,ϕS(A)〉 ∈ Cclq.

2. WheneverM is an L6∼-structure, A ∈ C6∼ and A 66∼M, the substruc-ture induced on the set A by 〈M,ϕS(M)〉 is exactly 〈A,ϕS(A)〉.

3. For every A ∈ C 6∼ and Bs ∈ Cclq such that 〈A,ϕS(A)〉 6s Bs, thereexists some C ∈ C6∼ such that A 66∼ C and Bs 6s 〈C,ϕS(C)〉.

Showing also that

(4) For any F ∈ C6∼, there exist A,B ∈ C6∼ with F 6 6∼ A,B such that A,Bare not isomorphic over F , but 〈A,ϕS(A)〉, 〈B,ϕS(B)〉 are isomorphicover F .

gives us, by Lemma 2.6.9, that R6∼(M 6∼) is not definable in 〈M 6∼, ϕS(M 6∼)〉.

We prove the statements in order.

58

Proof of (1). Let A ∈ C6∼. Denote As = 〈A,ϕS(A)〉. Clearly, As ∈ Cclq0 . Let

B ⊆ A be an arbitrary substructure of A. Denote by Bs the substructureinduced on B by As. Consider B, the substructure of A induced on the set

B = B ∪ {y1, . . . , yn−1 | R6∼(y1, . . . , yn−1,B) ∈ M(Bs)}Then

δ(B) = (|B|+ |B \B|)− |R6∼(B)|

≤ |B|+ (n− 1)|M(Bs)| −∑

K∈M(Bs)

|K|

= |B| −∑

K∈M(Bs)

|K|∗

= λ(Bs)Thus, λ(Bs) ≥ 0 and As ∈ Cclq.

Proof of (2). Let M, A be as in the statement.Let a1, . . . , an ∈ A be distinct and let {y1, . . . , yn−1} = Y ⊆ M be such

that M |=∧i≤nR6∼(y1, . . . , yn−1, ai). It must be that Y ⊆ A, for otherwise

δM(Y/A) ≤ (n− 1)− n < 0 in contradiction to A 66∼M.Thus, a ∈ ϕS(M) if and only if a ∈ ϕS(A).

Proof of (3). Let A ∈ C6∼, Bs ∈ Cclq be such that 〈A,ϕS(A)〉 6s Bs. DenoteAs = 〈A,ϕS(A)〉, by (1) we know As ∈ Cclq. Denote by B the universe ofBs.

For each K ∈ M(As) let yK = (y1K , . . . , y

n−1K ) ∈ An−1 be the unique tuple

such that R(yK , A) = K, and let K ∈ M(Bs) be the unique maximal clique

in Bs such that K ⊆ K. Let

R0 =⋃

K∈M(As)

{(y1K , . . . , y

n−1K , b) | b ∈ K \K}

For each L ∈ M(Bs) such that L ∩ A /∈ M(As), let zL1 , . . . , zLn−1 be new

distinct elements. Let

R1 =⋃

L∈M(Bs)L∩A/∈M(As)

{(zL1 , . . . , zLn−1, b) | b ∈ L}

Z =⋃

L∈M(Bs)L∩A/∈M(As)

{zL1 , . . . , zLn−1}

59

Define C to be the L 6∼-structure with universe B ∪ Z and

R6∼(C) = R6∼(A) ∪R0 ∪R1

and denote Cs = 〈C,ϕS(C)〉. Clearly Bs is a substructure of Cs. Moreover, asM(Cs) = M(Bs), we have that Bs 6s Cs.

We prove A 6 6∼ C, which also shows C ∈ C6∼. Let A ⊆ X ⊆ C. We haveto show that δC(X/A) ≥ 0.

Note that δC(X ∪ K/A) ≤ δC(X/A) for every K ∈ M(As). Also notethat whenever L ∈ M(Bs) with L ∩ X /∈ M(As) and |L ∩ X| ≥ n, thenδC(X ∪ {zL1 , . . . , zLn−1}/A) ≤ δC(X/A). Thus, it will suffice to prove the in-

equality under the assumption that K ⊂ X for every K ∈ M(As), and{zL1 , . . . , zLn−1} ⊆ X for every L ∈ M(Bs) with L∩X /∈ M(As) and |L∩X| ≥n. Then

δC(X/A) = (|X \ (X ∩B)|+ |(X ∩B) \ A|)− (|R0|+ |R1 ∩Xn|)

≥ |(X ∩B) \ A| −∑

K∈M(As)

|K \K| −∑

L∈M(Bs)|L∩X|≥n

L∩X/∈M(As)

|L ∩X|∗

= λBs(X ∩B/A) ≥ 0

Proof of (4). Let F ∈ C6∼. Define A, B to be the L 6∼-structures with universeF ∪ {a1, . . . , an} and

R6∼(A) = R6∼(F)

R6∼(B) = R6∼(F) ∪ {(a1, . . . , an)}

4.3 The pregeometry of Mclq

We show, using Proposition 2.5.3, that PG(Mclq) ∼= PG(M 6∼).

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Observation 4.3.1. For a structure A ∈ C∼, let As be the LS-structurewith universe A and S(As) = R∼(A). Then if there are no cliques in A ofsize greater than n, then λ(As) = δ∼(A). Thus, if there are no cliques in Aof size greater than n, then PG(A) = PG(As).Lemma 4.3.2. (C6∼, δ6∼) (Cclq, λ)

Proof. The proof is identical to the proof of 3.1.16 for the case g = Sn, usingthe above observation. Note that in the proof, the L∼-structure D has nocliques of size greater than n.

Lemma 4.3.3. (Cclq, λ) (C6∼, δ6∼)

Proof. Let As ∈ Cclq, A6∼ ∈ C6∼ with PG(As) = PG(A6∼) and universe A.Let Bs ∈ Cclq with As 6 Bs and universe B. Let f : M(Bs)→ Bn−1 be suchthat s(f(K)) ∈ [K]n−1 for every K ∈ M(Bs), and moreover s(f(K)) ∈ [A]n−1

whenever K ∩ A ∈ M(As). Define

R0 = {f(K)b | K ∈ M(Bs), K ∩ A ∈ M(As), b ∈ K \ A}R1 = {f(K)b | K ∈ M(Bs), K ∩ A /∈ M(As), b ∈ K \ s(f(K))}

and note that each r ∈ R0∪R1 has a unique K ∈ M(Bs) such that s(r) ⊆ K.Let B6∼ be the structure with universe B and

R6∼(B6∼) = R6∼(A6∼) ∪R0 ∪R1.

For any X ⊆ B, λBs(X/X∩A) ≤ δB6∼6∼ (X/X∩A). This is, informally, because

each membership of an element of X \ A in a relation in XB6∼ , implies themembership of this element in a clique in XBs . So A6∼ 66∼ B6∼ and B6∼ ∈ C6∼.

Say that a set X is good if for every K ∈ M(Bs), it holds that f(K) ∈Xn−1 if and only if |K ∩X| ≥ n. Then for a good X,

λBs(X/X ∩ A) = |X \ A| −∑

K∈MBs (X)K∩A∈M(As)

|K \ A| −∑

K∈MBs (X)K∩A/∈M(As)

|K|∗

= |X \ A| − |R0 ∩ (Xn \ An)| − |R1 ∩ (Xn \ An)|

= δB6∼6∼ (X/X ∩ A).

By 2.5.4, in order to show PG(Bs) = PG(B6∼) it suffices to show thatwhenever Y ⊆ B is a set such that clBs(Y ) = Y or clB 6∼(Y ) = Y , then

λBs(Y/Y ∩ A) = δB6∼6∼ (Y/Y ∩ A). As every such Y is good, we are done.

Theorem 4.3.4. PG(Cclq, λ) ∼= PG(C6∼, δ6∼)

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4.4 A two-sorted variation of the construc-

tion

In Definition 4.1.3 we restrict the class Cclq0 by requiring that two distinct

maximal cliques can intersect in a set of size strictly smaller than n, wheren is the arity of the unique relation in the language L. This is a technicalrequirement simplifying the construction: if we allow cliques to intersect inarbitrarily large sets uniqueness of the standard amalgam is lost and – moreimportantly – the intersection of two strong sub-structures need no longeritself be strong:

Example. For i = 1, 2, let Ki = {a1, . . . , an, bi1, b

i2}. Let D be the LS-

structure with universe K1 ∪ K2 and M(D) = {K1, K2}. For j = 1, 2 letBj = {a1, . . . , an, b

1j , b

2j}. Then Bj 6s D with λD(Bj) = (n + 2) − (2 + 2) =

n − 2. However, denoting A = B1 ∩ B2, we have A 66s D. This is becauseA ∩K1 = A ∩K2 and therefore MD(A) = {A}, resulting in λD(A) = n− 1.

These problems can, apparently, be dealt with by changing the class ofstrong embeddings. We sketch a different approach – focusing on the role ofimaginary elements in the construction. We work in a two-sorted languageL+ with sorts Sv, Sc and a single binary relation e on Sv × Sc. We think ofthe elements of Sv as vertices, and of elements of Sc as names of maximalcliques. For an L+ structure A, denote by ASv the set of its elements of sortSv, and by ASc the set of its elements of sort Sc.

Let A be an L+-structure. For every c ∈ ASc , define the set c = {v ∈ASv : A |= e(v, c)}. If A is finite, define

λ(A) = |ASv | −∑c∈ASc

|c|∗.

The function λ is submodular. Explicitly, let D be any L+-structure andA,B ⊆ D substructures. Define the structure A∪B as follows: (A∪B)Sv =ASv ∪BSv , (A∪B)Sc = ASc ∪BSc , (A∪B)e = De ∩ ((A ∪ B)Sv × (A ∪ B)Sc).Define the structure A ∩ B similarly. Then

λ(A ∪ B) + λ(A ∩ B) ≤ λ(A) + λ(B).

We define the strong substructure relation, 6+, on L+-structures as usual.Define C+ to be the class of finite L+-structures A with ∅ 6+ A.

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Observation. If A,B ∈ C+ and A 6 B, then |c ∩ ASv | < n for everyc ∈ BSc \ ASc .

Definition. Let B1,B2 ∈ C+ and A = B1 ∩ B2 a common substructure suchthat A 6 B1,B2. Assume (by renaming elements of BSc1 if necessary) that|{v ∈ ASv : A |= e(v, c)}| ≥ n for every c ∈ BSc1 ∩ BSc2 . Let D := B1 q B2 bethe L+-structure with

DSv = BSv1 ∪ BSv2 ,DSc = BSc1 ∪ BSc2 ,De = Be1 ∪ Be2.

Then as before B1,B2 6 D and D ∈ C+. Moreover λ(D/B1) = λ(B2/A).So C+ is an amalgamation class whose generic structure we denote M+.The analysis from this point on goes unaltered from the one in the previoussections of this chapter.

Remark. In the above definition, the assumption that B1 and B2 do notshare cliques that are small in A is necessary for obtaining an amalgam inC+. Indeed, consider the case BSvi = {a, bi1, . . . , bin}, BSci = c, Bie = BSvi ×{c}.We get λ(B1 q B2/B1) = 0 but λ(B2/B1 ∩ B2) = n− 2.

To every L+-structure A, we associate the LS-structure As with universeASv and M(As) = {c | c ∈ ASc , |c| ≥ n}. In general λ(A) ≤ λ(As), wherestrict inequality occurs whenever there are distinct c, d ∈ ASc such thatc = d ∈ M(As). In other words, L+ tells us in advance whether a clique inLS will eventually split into several cliques or not.

Finally, we observe that we can recover M+ from M+s using imaginary

elements. Indeed, there exists k ∈ N (depending on n) such that two distinctcliques of M(M+

s ) cannot intersect at more than k elements1.For k-tuples of elements of M+

s , write (a1, . . . , ak) ∼ (b1, . . . , bk) if

1. The elements a1, . . . , ak are distinct and the elements b1, . . . , by aredistinct.

2. For every element x, the set {a1, . . . , ak, x} is a clique if and only if theset {b1, . . . , bk, x} is a clique.

1If K1,K2 are cliques intersecting in k elements then λ(K1 ∪K2) ≤ (|K1|+ |K2|−k)−(|K1| − (n− 1) + |K2| − (n− 1)) = 2n− 2− k, so any k > 2(n− 1) will do.

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Then ∼ is a ∅-definable equivalence relation, where each non-singleton equiv-alence class corresponds to a unique element of M(M+

s ), in turn correspondingto an element c ∈ (M+)Sc . For each imaginary element c ∈ M+

s

/∼ and ele-

ment v ∈M+s , let e(v, c) if there exist x1, . . . , xk−1 such that (v, x1, . . . , xk−1) ∈

c. Thus, we have recovered M+.

4.5 The geometric construction

For a pregeometry (G, cl) observe that 〈G,Dn〉 (where Dn is the set of de-pendent n-tuples in X) is an LS-structure in Cclq

0 . Say that an LS-structureA is geometric if

• Whenever X ⊆ A with |X| ≥ n and λ(X) < n, then there exists aunique K ∈ M(A) with X ⊆ K.

Observation 4.5.1. If A is a geometric LS-structure and X ⊆ A is suchthat |X| ≥ n and λA(X) < n, then X is a clique and λA(X) = n− 1. Thus,X 6s A for any X ⊆ A with |X| < n.

Definition 4.5.2. Define Cgeo to be the class of geometric LS-structures.

Observation 4.5.3. The class Cgeo is a subclass of Cclq.

Definition 4.5.4. Let A1,A2 ∈ Cgeo and let B = A1 ∩ A2 be a commoninduced substructure. Define the geometric amalgam of A1 and A2 overB to be the unique LS-structure D whose universe is A1 ∪ A2 such thatM(D) = M ∪M1 ∪M2 where

M = {K1 ∪K2 : K1 ∈ M(A1), K2 ∈ M(A2), |K1 ∩K2| ≥ n− 1}M1 = {K ∈ M(A1) : ∀L ∈ M(A2) |K ∩ L| < n− 1}M2 = {K ∈ M(A2) : ∀L ∈ M(A1) |K ∩ L| < n− 1}

The above amalgam differs from the standard amalgam in the definitionof M , where we take the union of cliques if they overlap at n−1 points ratherthan in n points.

Lemma 4.5.5. Let A1,A2 ∈ Cgeo with B = A1 ∩ A2 and B 6s A1. Let Dbe the geometric amalgam of A1 and A2 over B. Then D ∈ Cgeo.

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Proof. Let X ⊆ D with |X| ≥ n and λD(X) < n. We must show there is aunique K ∈ M(D) such that X ⊆ K. Since A1,A2 ∈ Cgeo, we may assumethat X * A1 and X * A2.

We show that there is at most one clique containing X. Assume there areK,L ∈ M(D) distinct such that X ⊆ K∩L. Then K = K1∪K2, L = L1∪L2

where Ki, Li ∈ M(Ai) and it must be that |K1 ∩ B|, |L1 ∩ B| ≥ n − 1. LetC = (K1∪L1)∩B, let x ∈ X \A2. Then λD(C∪{x}/C) < 0 in contradictionto B 6s A1.

We have left to show that X is contained in a clique. It suffices to showthat this holds of some superset of X, so we may enlarge X. Thus, assumethat for any K ∈ M(D), if |K ∩ X| ≥ n, then K ⊆ X. This impliesλD(X/X ∩ A2) = λD(X ∩ A1/X ∩B).

We claim |X ∩ A1| ≥ n. Assume |X ∩ A1| < n. Then λD(X/X ∩ A2) =|X∩(A1\A2)|. It must be that |X∩A2| ≥ n, for otherwise λD(X) = |X| ≥ n,in contradiction. As λD(X ∩ A2) < λD(X) < n and |X ∩ A2| ≥ n, this is acontradiction to A2 being geometric. A symmetric argument yields that also|X ∩ A2| ≥ n.

Now, since B 6s A1, we have that λD(X/X ∩ A2) ≥ 0. Thus, λD(X ∩A2) ≤ λD(X) < n. Since |X ∩ A2| ≥ n, by Observation 4.5.1, there existsK2 ∈ M(A2) such that X ∩ A2 = K2 and λD(X ∩ A2) = n − 1. As λD(X ∩A1/X∩B) = λD(X/X∩A2) = 0 and, sinceX∩B ⊆ K2, also λ(X∩B) ≤ n−1,we have λD(X ∩ A1) < n. Therefore, there is some clique K1 ∈ M(A1) suchthat X ∩ A1 = K1. If |K1 ∩K2| < n− 1, then λD(X) ≥ n in contradiction.So |K1 ∩K2| ≥ n− 1 and K1 ∪K2 ∈ M(D) contains X.

Remark 4.5.6. The geometric amalgam is also an amalgam for Cclq.

Proposition 4.5.7. (Cgeo, λ) is a simple Fraısse-Hrushovski amalgamationclass.

Proof. Clearly Cgeo is closed under isomorphism and substructures. By theabove lemma, similarly to the proof of Lemma 4.1.11, we have that thegeometric amalgam is a simple amalgam for (Cgeo, λ).

Notation. Denote the generic structure of (Cgeo, λ) by Mgeo.

Observation 4.5.8. The set of closed sets of dimension (n−1) in PG(Mgeo)is exactly M(Mgeo).

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4.6 Mgeo is the pregeometry of a generic struc-

ture

In this final section, we show that Mgeo is essentially the pregeometry of ageneric structure.

Definition 4.6.1. Say that a pregeometry (G, cl) with dimension functiondG is flat if whenever E1, . . . , En are closed sets in G, then∑

s⊆{1,...,k}

(−1)|s|dG(Es) ≤ 0

where E∅ = cl(⋃ni=1En) and Es =

⋂i∈sEi for every ∅ 6= s ⊆ {1, . . . , n}.

Definition 4.6.2. We say that a pregeometry (X, cl) is n-pure if n is themaximal natural number such that every n-tuple of X is independent.

Notation. For any pregeoemtry A, we denote its closure operator by clAand its dimension function by dA.

Definition 4.6.3. Let A be a flat, (n − 1) pure pregeometry with universeA. We define Ageo to be the LS-structure with universe A and M(Ageo) ={clA(B) | B ∈ [A]n−1, clA(B) 6= B}.

Observation 4.6.4. If B ⊆ A, then Bgeo ⊆ Ageo.

Observation 4.6.5. For every A ∈ Cgeo,

• PG(A) is (n− 1)-pure.

• (PG(A))geo = A.

Notation. For every structure A ∈ Cclq ∪C∼ ∪C6∼, denote A := PG(A)geo.

Lemma 4.6.6. For any flat, (n− 1)-pure pregeometry B and A ⊆ B finite,we have λ(Ageo) ≥ dB(A).

Proof. Assume |A| > n− 1, for otherwise λ(Ageo) = dB(A) = |A|. Let

A = A ∪⋃

X∈[A]n−1

clB(X).

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Clearly λ(Ageo) ≥ λ(Ageo) and dB(A) ≥ dB(A), so we may assume A = A.Letting A =

⋃M(Ageo), note that λ(Ageo) − λ(Ageo) = |A \ A| ≥ dB(A) −

dB(A), so we may also assume A = A.Enumerate M(Ageo) = {E1, . . . , Ek} and note that A =

⋃ki=1Ei. We use

the notation of Definition 4.6.1. Observe that by (n−1)-purity, dB(Es) = |Es|whenever |s| ≥ 2. Then

∑∅6=s⊆{1,...,k}

(−1)|s|+1dB(Es) =k∑i=1

dB(Ei) +∑

s⊆{1,...,k}|s|≥2

(−1)|s|+1|Es|

=k∑i=1

|Ei| − |Ei|∗ +∑

s⊆{1,...,k}|s|≥2

(−1)|s|+1|Es|

=∑

∅6=s⊆{1,...,k}

(−1)|s|+1|Es| −k∑i=1

|Ei|∗

= |k⋃i=1

Ei| −k∑i=1

|Ei|∗

= |A| − s(Ageo) = λ(Ageo)

By flatness, −dB(A) +∑∅6=s⊆{1,...,k}(−1)|S|+1dB(Es) ≥ 0, so λ(Ageo) ≥ dB(A).

Lemma 4.6.7. Let A ∈ Cclq be such that PG(A) is (n − 1)-pure, then

A ∈ Cgeo.

Proof. By the above lemma, we have that A ∈ Cclq. Let X ⊆ A with |X| ≥ n

and λA(X) < n such that λA(X) is minimal. Note that by our choice of X,

it must be that⋃

MA(X) = X. We may assume MA(X) ⊆ M(A), replacing

each clique with the maximal clique containing it can only lower λA(X).

Enumerate MA(X) = {K1, . . . , Kr}. We show inductively that for every

m ≤ r, for the set Xm =⋃mi=1Ki we have λA(Xm) ≥ λA(Xm). Assume this

holds for m.

Case 1: |Km+1 ∩Xm| ≥ n− 1, then λA(Xm+1/Xm) = 0. By submodularity,

λA(Xm+1/Xm) ≤ λA(Km+1/Km+1 ∩Xm).

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We know λA(Km+1∩Xm) ≥ n−1 and λA(Km+1) = n−1, so λA(Xm+1/Xm) ≤0.

Case 2: |Km+1 ∩Xm| < n− 1. Then

λA(Xm+1/Xm) = |Km+1 \Xm| − |Km+1|∗= |Km+1| − |Km+1 ∩Xm| − (|Km+1| − (n− 1))

= (n− 1)− |Km+1 ∩Xm|.

On the other hand, by submodularity

λA(Xm+1/Xm) ≤ λA(Km+1/Km+1 ∩Xm)

= λA(Km+1)− λA(Km+1 ∩Xm)

= (n− 1)− |Km+1 ∩Xm|.

In any case, λA(Xm+1/Xm) ≥ λA(Xm+1/Xm) and λA(Xm) ≥ λA(Xm), so

λA(Xm+1) ≥ λA(Xm+1).

As X = Xr we have λA(X) ≤ λA(X) < n. So X ⊆ clA(X) ∈ M(A).

Lemma 4.6.8. For every Aclq ∈ Cclq for which PG(Aclq) is (n−1)-pure thereis some A∼ ∈ C∼ such that PG(Aclq) = PG(A∼). Moreover, if Aclq 6s Bclq

for some Bclq ∈ Cclq, then there is B∼ ∈ C∼ with A∼ 6s B∼ and PG(B∼) =PG(Bclq).

Proof. Repeat the proof of 4.3.3, defining R0, R1 as sets of symmetric tuples.Since PG(Aclq) is (n−1)-pure, no pair of cliques of M(Aclq) intersect in n−1elements, and the proof goes through.

The following lemma is an adaptation of [EF11, Second Changing Lemma],with the same proof.

Lemma 4.6.9. Let A,D ∈ C∼ with A 6∼ D. Let B ∈ Cclq with PG(B) =PG(A). Let D′ be the structure with the same universe as D, and S(D′) =(R∼(D) \R∼(A)) ∪ S(B). Then B 6s D′ and PG(D′) = PG(D).

For the next lemma only, we think of elements A ∈ C∼ as LS-structureswith S(A) = R∼(A), and we do so implicitly. Observe that δ∼ and λ coincideon a structure of C∼ and its LS counterpart.

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Lemma 4.6.10. Let A ∈ Cclq be such that PG(A) is (n − 1)-pure. Let

B ∈ Cgeo be such that A 6s B. Then there exists D ∈ Cclq such thatA 6s D, PG(D) is (n− 1)-pure, and D = B.

Proof. By Lemma 4.6.8, let A∼ ∈ C∼ be such that PG(A∼) = PG(A) andlet B∼ ∈ C∼ be such that A∼ 6s B∼ and PG(B∼) = PG(B). Let D bethe structure obtained from B∼ by replacing A∼ with A, as in Lemma 4.6.9.Then A 6s D, D ∈ Cclq, and PG(D) = PG(B∼) = PG(B). By Observation

4.6.5, PG(D) is (n− 1)-pure and D = B.

Definition 4.6.11. Let C∼n be the class of structures A ∈ C∼ such thatPG(A) is (n− 1)-pure.

The class (C∼n , δ∼) is a simple Fraısse-Hrushovski amalgamation classwith respect to free amalgamation. Denote by M∼n the generic structureof (C∼n , δ∼).

The following is a special case of Lemma 15 of [Hru93].

Lemma 4.6.12. Let A ∈ C∼n . Then λ(A) = dA(A).

Proof. Enumerate M(A) = {E1, . . . , Ek}. We use the notation of Definition4.6.1.

Note that R∼[A] =⋃ki=1 R

A∼[Ei], because whenever (a1, . . . , an) ∈ R∼[A],

then cl({x1, . . . , xn}) ∈ M(A). Then E∅ 6 A and λ(A)−λA(E∅) = |A\E∅| =dA(A) − dA(E∅), and so we may assume A = E∅. Observe that Es 6∼ Afor any s 6= ∅, as an intersection of closed sets. Recall that in the proof oflemma 4.6.6 we saw that

λ(A) =∑

∅6=s⊆{1,...,k}

(−1)|S|+1dA(Es).

Then

λ(A) =∑

∅6=s⊆{1,...,k}

(−1)|s|+1dA(Es)

=∑

∅6=s⊆{1,...,k}

(−1)|s|+1δA∼(Es)

=∑

∅6=s⊆{1,...,k}

(−1)|s|+1|Es| −∑

∅6=s⊆{1,...,k}

(−1)|s|+1|RA∼[Es]|

= |A| − |R∼[A]| = δ∼(A) = dA(A)

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where the equality between the third and fourth line is by the inclusion-exclusion principle.

Proposition 4.6.13. M∼n ∼= Mgeo

Proof. By generalizing the method of 2.6.7, we need only show that:

• If A ∈ C∼n , then A ∈ Cgeo.

• If A 6∼ B ∈ C∼n , then A ⊆ B.

• If A ∈ C∼n , B ∈ Cgeo such that A 6s B, then there exists A 6∼ C ∈ C∼nsuch that B 6s C.

The first point is Lemma 4.6.7, the second point is because A 6∼ Bimplies PG(A) ⊆ PG(B), the third point follows from Lemma 4.6.10.

Lemma 4.6.14. Whenever A 6∼ M∼n , then A 6s M∼n . Additionally,

δ∼(A) = λ(A).

Proof. For the first part, apply 2.6.5. For the additional part, observe thatPG(A) ⊆ PG(M∼n ). Then by Lemma 4.6.12, we have

λ(A) = dPG(A)(A)

= dPG(M∼n )(A)

= δ∼(A).

Note that the following theorem is about equality of pregeometries, andnot mere isomorphism.

Theorem 4.6.15. Identifying M∼n withMgeo, we have PG(M∼n ) = PG(Mgeo).

Proof. Let A ⊆M∼n be finite.Let B 6∼ M∼n be the self sufficient closure of A in M∼n . Then by the

above lemma B 6s Mgeo and

dPG(M)(A) = δ∼(B) = λ(B) ≥ dPG(Mgeo)(A).

Let D 6s Mgeo be the self sufficient closure of A in Mgeo. Let D ⊆M∼n be the

structure induced by M∼n on D, the underlying set of D. By Lemma 4.6.6we have

dPG(M)(A) ≤ dPG(M)(D) ≤ λ(D) = dPG(Mgeo)(A).

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The above theorem shows that Mgeo is infact the reduct of PG(M∼n ) =PG(Mgeo) to Dn. So clearly, the structure of Mgeo can be read off the prege-ometry of Mgeo and vice versa.

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Chapter 5

Closed ordinal Ramseynumbers

5.1 Introduction

For a set of ordinals J , denote the order-type of J by ord(J). For an ordinalα, denote [J ]α = {X ⊆ J | ord(X) = α}. For a nonzero cardinal κ, a naturalnumber n, and ordinals β and αi for all i ∈ κ, we write

β →cl (αi)ni∈κ

to mean that for every colouring c : [β]n → κ of subsets of β of size n in κmany colours, there exist some i ∈ κ and X ⊆ β such that ord(X) = αi, X isclosed in its supremum, and [X]n ⊆ c−1({i}). Should such an ordinal exist,let Rcl(αi)

ni∈κ denote the least ordinal β for which the statement β →cl (αi)

ni∈κ

holds. We call Rcl(αi)ni∈κ the closed ordinal Ramsey number of (αi)

ni∈κ. When

n is omitted, by convention n = 2.For a history of partition relations and Rado’s arrow notation see [HL10].

The ordinal partition calculus was introduced by Erdos and Rado in [ER56].Topological partition calculus was considered by Baumgartner in [Bau86].Baumgartner’s work was continued in recent papers on topological (closed)ordinal partition relations by Caicedo, Hilton, and Pina [Pn15], [Hil16],[CH17].

Caicedo and Hilton proved recently that ω2 · 3 ≤ Rcl(ω · 2, 3) ≤ ω3 · 100[CH17, Theorem 8.1] and provided also the upper bound Rcl(ω2, k) ≤ ωω forevery positive integer k [CH17, Theorem 7.1]. The lower bound Rcl(ω2, k) ≥ωk+1 is a consequence of [CH17, Theorem 3.1].

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In this thesis, we calculate the exact value Rcl(ω · 2, 3) = ω3 · 2.

5.1.1 Description of proof

In order to prove Rcl(ω · 2, 3) ≤ ω3 · 2, we must show that for each colouringc : [ω3 · 2]2 → 2 with no homogeneous triples of colour 1, there is somehomogeneous Y ∈ [ω3 · 2]ω·2 of colour 0, closed in its supremum. However,the first part of the proof is the canonization of any finite pair-colouring ofan ordinal less than ωω.

We show first that for each δ < ωω, k ∈ N and pair-colouring c : [δ]2 → k,there exists some X ⊆ δ, closed in its supremum with ord(X) = δ, such thatc � [X]2 is a canonical colouring (Definition 5.3.12). A canonical colouringis one such that for “most” {α, β} ∈ [δ]2, the value c({α, β}) depends onlyon the Cantor-Bendixson ranks of the points α, β in the topological space δ,and on the Cantor normal form of δ.

The bound Rcl(ω · 2, 3) ≤ ω3 · 2 is achieved by a combinatorial analysis ofcanonical colourings in two colours, and the following simple lemma on finitesets:

Lemma 5.1.1. Whenever F ⊆ Fin(N) is an infinite ⊆-antichain, thereexists an infinite set {Ai : i < ω} ⊆ F and points {ki : i < ω} ⊆ N such thatki ∈ Ai \

⋃j 6=iAj.

Proof. First, for every countably infinite collection {Ai : i < ω} of distinctfinite subsets of N, there is some infinite I ⊆ ω and sets {Xi : i ∈ I} suchthat Ai ∩ Aj = Xi for all i, j ∈ I such that i < j: we define sets Bn andnatural numbers in such that

1. B0 = ω

2. in = minBn

3. Bn+1 ⊆ {i ∈ Bn | i > in} is infinite and Ain ∩ Aj is constant for allj ∈ Bn+1.

Point 3 is gotten by applying the pigeonhole principle, as Ain is finite andBn is infinite. Thus Xin =

⋂k≥nAik , and hence m < n =⇒ Xm ⊆ Xn.

Given an infinite ⊆-antichain F ⊆ Fin(N), we may assume by shrinkingit, that it is as above for some enumeration F = {Ai}i∈ω. Let Yi = Ai \Xi.Since F is an antichain, Yi 6= ∅. Moreover, for distinct i, j it holds that

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Yi ∩ Aj ⊆ Xmin{i,j} ⊆ Xi. This implies that Yi is disjoint from Aj for everyi 6= j. Now choose ki ∈ Yi arbitrarily.

The bound Rcl(ω ·2, 3) ≥ ω3 ·2 is achieved by describing a single colouringc : [ω3·2]2 → 2 such that for each θ < ω3·2, the colouring c � [θ]2 demonstratesRcl(ω · 2, 3) > θ.

5.2 Preliminaries

For every nonzero ordinal α there exist a unique nonzero k ∈ N and a uniquesequence of ordinals β1 ≥ ... ≥ βk such that

α = ωβ1 + ωβ2 + · · ·+ ωβk .

Also, there exist a unique l ∈ N, a sequence of ordinals γ1 > · · · > γl, and asequence of nonzero natural numbers m1, . . . ,ml such that

α = ωγ1 ·m1 + ωγ2 ·m2 + · · ·+ ωγl ·ml.

These representations of α are two variations of the Cantor normal form ofan ordinal α. Here, we shall refer to the first as the Cantor normal form.The Cantor-Bendixson rank1 (CB rank) of α is βk (which is equal to γl), andis denoted by CB(α). We also denote L(α) = ml. For the ordinal α = 0, seethe following remark.

Remark 5.2.1. We wish to consider CB(α) and L(α) for every ordinal α,but these values do not naturally make sense for 0. We solve this for α = 0by either omitting 0 completely, or arbitrarily setting CB(0) = 0 and L(n) =n+ 1 for each n ∈ ω, virtually omitting 0.

Beside the ordinary ≤ order relation on ordinals, we endow the ordinalswith an anti-tree ordering <∗⊆≤. By anti-tree, we mean that {β | α <∗ β}is well-ordered by <∗ for all α. This is the same ordering <∗ presented insection 8 of [CH17].

We say that β <∗ α whenever α = β + ωγ for some nonzero ordinal γwith γ > CB(β). Equivalently, for some γ > CB(β), α is the least ordinalof CB rank γ with β ≤ α. We write β C∗ α if α is the unique immediatesuccessor of β in <∗. A graphical representation of <∗ on ω3 · 2 can be foundin Figure 5.2 at the end of this Chapter.

1This is the Cantor-Bendixson rank of α as a point in the space of ordinals

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Notation 5.2.2. Let α be some ordinal.

• Denote T(α) = {α}∪{β | β <∗ α} and T=n(α) = {β <∗ α | CB(β) = n}.

• Assuming CB(α) = γ + 1 is a successor ordinal, denote

Fan−(α) = {β | β C∗ α}.

Equivalently, Fan−(α) = {α + ωγ · (i+ 1) | i ∈ ω}.

• Letting α be the unique ordinal such that αC∗ α, denote

Fan(α) = Fan−(α),

the unique set of the form Fan−(β) of which α is a member.

• For ordinals α, β denote by α∨ β the least ordinal γ such that α, β <∗ γ.

Let I be a set of ordinals.

• Denote by ρI the unique order preserving bijection from I onto ord(I).

• Write J ⊆cof I to mean that J is a cofinal subset of I.

From here onwards, all ordinals are assumed to be smaller thanωω.

5.2.1 Skeletons

The following is the main notion we use in thinning out arguments.

Definition 5.2.3. Let δ < ωω be an ordinal. We say that I ⊆ δ is a skeletonof δ if:

(S1) I is closed in δ in the order topology;

(S2) ord(I) = δ;

(S3) α <∗ β if and and only if ρI(α) <∗ ρI(β), for all α, β ∈ I.

For an arbitrary set of ordinals J , we say that I ⊆ J is a skeleton of J ifρJ [I] is a skeleton of ord(J). Write I ⊆sk J .

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Definition 5.2.4. For each n ∈ N, write I ⊆skn J and say that I is an

n-skeleton of J if I ⊆sk J and

{α ∈ J | Fan(α) ∩ I 6= ∅ and L(ρJ(α)) ≤ n} ⊆ I.

Informally, if a fan was not removed entirely in the transition from J to I,then its first n elements were not removed. Note that the relation ⊆sk

0 is infact ⊆sk and that ⊆sk

m implies ⊆skn whenever m > n.

Observation 5.2.5. For every natural n, the relation ⊆skn is transitive.

Definition 5.2.6. Let δ be an ordinal with Cantor normal form

δ = ωγ1 + ωγ2 + · · ·+ ωγk .

For β < δ define CNFδ(β) to be the least i such that β ≤ ωγ1 + · · ·+ ωγi .Define CC δ(i) = {β < δ | CNFδ(β) = i}. Explicitly, CC δ(i) is the collectionof β < δ such that

∑i−1j=1 ω

γj < β ≤∑i

j=1 ωγj .

Define kδ to be the number of non-empty components of δ. That is, kδ = kif δ is a limit ordinal, and kδ = k − 1 if δ is a successor ordinal.

The following observation is a straightforward unravelling of the defini-tions. Note that by definition ρI is a <∗-isomorphism.

Observation 5.2.7. If I ⊆sk δ for an ordinal δ < ωω, then for every α ∈ Iwe have CNFδ(α) = CNFδ(ρI(α)) and CB(α) = CB(ρI(α)).

For the remainder of this section, fix some ordinal δ < ωω.

Observation 5.2.8. The collection {CC δ(i) | 1 ≤ i ≤ kδ} is a partition of δinto kδ closed subsets of δ. Moreover, CC δ(kδ) is of order type ωγkδ , and forevery i < kδ, the set CC δ(i) is either a singleton (when γi = 0) or of ordertype ωγi + 1.

The following lemma follows directly from the definitions and the unique-ness of the Cantor normal form.

Lemma 5.2.9. Let I ⊆ δ. Then I ⊆skn δ if and only if I∩CC δ(i) ⊆sk

n CC δ(i)for every 1 ≤ i ≤ kδ.

The following is a way of ensuring an infinite successive thinning out toskeletons is a skeleton.

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Definition 5.2.10. Let γ be some ordinal. For each i < γ, let Ii ⊆ δ. Wesay that the collection {Ii : i < γ} is fan preserving if whenever α ≤ γ is alimit ordinal and β < δ is such that Fan(β) ∩ Ii is infinite for all i < α, thenFan(β) ∩

⋂i<α Ii is infinite.

Lemma 5.2.11. Let k be some positive integer. Let {Ii : i < γ} be afan preserving collection of n-skeletons of ωk + 1. Then I =

⋂i<γ Ii is an

n-skeleton of ωk + 1.

Proof. By induction on k. Every skeleton of ωk + 1 must contain the pointωk and so ωk ∈ I. Additionally, every skeleton intersects infinitely withFan−(ωk). By fan preservation, Fan−(ωk) ∩ I is infinite, and contains thefirst n elements of Fan−(ωk).

Enumerate Fan−(ωk) ∩ I = {βi : i < ω}. For every α < γ, we haveT(βi)∩Iα ⊆sk

n T(βi). The collection {T(βi)∩Iα}α<γ of n-skeletons of T(βi) ∼=ωk−1 + 1 is fan preserving. By the induction hypothesis T(βi) ∩ I is an n-skeleton of T(βi). Thus,

I =⋃i∈ω

(T(βi) ∩ I) ∪ {ωk}

is an n-skeleton of ωk + 1.

Corollary 5.2.12. Let γ be some ordinal. Whenever {Ii : i < γ} is a fanpreserving collection of n-skeletons of δ, then I =

⋂i<γ Ii is an n-skeleton of

δ.

Lemma 5.2.13. Let {Ii : i < γ}, for some ordinal γ, and assume that thereis some σ : γ → N with finite fibers (that is |σ−1[{n}]| <∞ for each n ∈ N)such that Iα ⊆sk

σ(α)

⋂i<α Ii for all i < γ. Then {Ii : i < γ} is a fan preserving

collection of skeletons of I0.

Proof. By induction on γ. Let α ≤ γ be a limit ordinal such that {Ii}i<jis fan preserving for every j < α. By Corollary 5.2.12, Ij ⊆sk I0 for everyj < α.

Let β be such that Fan(β) ∩ Ii is infinite for all i < α. For an arbitraryn ∈ N, let in < α be maximal such that σ(in) ≤ n. By an inductionargument on j, Ij ⊆sk

n Iin for every in < j < α. Thus, |Fan(β) ∩ Iα| ≥ n forany n ∈ N.

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Figure 5.1: The sets F (ω3)1n

... ... ...

. . .

... ... ...

. . .

... ... ...

. . .

. . .

F (ω3)13

F (ω3)12

F (ω3)11

F (ω3)10

5.2.2 The sets F (ωk)rn

We will thin out colourings using families of skeletons as in the statement ofLemma 5.2.13. This compromise prevents us from controlling the colouringon entire sets of the form T=n(α), but it allows us to control it on “large”sets of a specific form.

Let k, r ∈ N. We recursively define a set F (ωk)rn ⊆ T=n(ωk) for eachn ≤ k (see Figure 5.1)

F (ωk)rn =

{{ωk} , n = k⋃β∈F (ωk)rn+1

{γ ∈ Fan−(β) | L(γ) > r} , n < CB(α)

An equivalent non-recursive definition, when n < k, is:

F (ωk)rn = {β ∈ T=n(ωk) | min{L(β′) : β <∗ β′ <∗ ωk} > r, L(β) > r}.

Note that F (ωk)0n = T=n(ωk).

We extend the definition to an arbitrary α < ωω. Denote k = CB(α).Observe that T(α) ∼= ωk + 1 and define

F (α)rn = ρ−1T(α)[F (ωk)rn]

Definition 5.2.14. Fix some α < ωω and n ≤ CB(α). Define

F(α)rn = {A ⊆ T=n(α) | F (α)rn ⊆ A}

F(α)n =⋃r∈N

F(α)rn.

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Example 5.2.15. As an instructive example, the set⋃k∈ω

{ω · k + l | l > k}

is not an element of F(ω2)0.

Observation 5.2.16. For a fixed n, the set F(α)n is closed under finiteintersections. This is due to the fact F(α)rn ⊆ F(α)sn, whenever r < s. Theset F(α)n is the filter on T=n(α) generated by {F (α)rn}r∈N.

Fact 5.2.17. Let δ < ωω be an ordinal, let I ⊆sk δ and let α ∈ I. ThenρI [F ∩ I] ∈ F(ρI(α))rn, for any F ∈ F(α)rn.

Fact 5.2.18. Let θ < ωω be some ordinal and let A ⊆ T=n(θ) for somen ≤ CB(θ). Then for any n ≤ k ≤ CB(θ), A ∈ F(θ)rn if and only if

{β ∈ T=k(θ) | T(β) ∩ A ∈ F(β)rn} ∈ F(θ)rk

Corollary 5.2.19. Let θ be some ordinal and let l1 < · · · < lk < CB(θ) andA1, . . . , Ak be such that Ai ∈ F(θ)li . Then there exists some X ⊆

⋃kn=1An

closed in its supremum with ord(X) = ωk. Moreover, X can be chosen suchthat ρX [X ∩ F ] ∈ F(ωk)i for any 1 ≤ i ≤ k and F ∈ F(θ)li .

Proof. By induction on k. Choose some B = {br : r < ω} with ord(B) = ωsuch that br ∈ F (θ)rlk . Let

Ci = {β ∈ T=lk(θ) | T(β) ∩ Ai ∈ F(β)li}.

By the above lemma Ci ∈ F(θ)k, and so we have that Ak ∩⋂k−1i=1 Ci ∈ F(θ)lk .

Denote this set by Ak. For each α ∈ Ak, by induction hypothesis, choosesome Xα ⊆

⋃k−1i=1 (T(α) ∩ Ai) closed in its supremum with ord(Xα) = ωk−1.

Then X =⋃α∈A∩BXα is closed in its supremum with ord(X) = ωk.

5.3 Reducing a finite pair-colouring to a canon-

ical colouring

For this section fix some ordinal δ < ωω and some positive integer n.

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Definition 5.3.1. Let c : δ → n be some colouring. For r ∈ N, say that c isr-good if for every θ < δ and l < CB(θ), there is some colour c < n such that

{α ∈ T=l(θ) | c(α) = c} ∈ F(θ)rl .

Say that I ⊆sk δ is r-good with respect to c, if c ◦ ρ−1I is r-good.

Lemma 5.3.2. Let c : δ → n be some colouring. Then for any r ∈ N thereexists some I ⊆sk

r δ r-good with respect to c.

Proof. By Lemma 5.2.9, we may assume δ = ωk+1 + 1 for some k ∈ N. Weprove by induction on k.

For i ∈ N, denote Ti := T(ωk · (i+ 1)). Considering c � Ti as a colouringof ωk + 1, by the induction hypothesis there is some r-good Si ⊆sk

r Ti. SinceSi is r-good, let fi : k + 1 → n be the function mapping m to the colourassigned to a large set of α ∈ Si with CB(α) = m. By the pigeonholeprinciple, there is an infinite subset B ⊆ N such that fi = fj for all i, j ∈ B.Let B′ = B ∪ N<r and take I =

⋃i∈B′ Si. By construction we have that

I ⊆skr ωk+1 and r-goodness follows by Fact 5.2.18.

Definition 5.3.3. Let c : [δ]2 → n be some colouring. Say that c is normal ifwhenever β1, β2 < δ, β1 <

∗ β2, the value c({β1, β2}) depends only on CB(β1),CB(β2) and CNFδ(β2). Namely, there is a function c, independent of β1, β2,such that

c({β1, β2}) = c(CNFδ(β2),CB(β2),CB(β1)).

Say that I ⊆sk δ is normal with respect to c if c ◦ ρ−1I is normal.

Notation 5.3.4. For a pair-colouring c : [X]2 → n and an element α ∈ X,write cα for any colouring cα : X → n assigning cα(β) = c({α, β}) to everyβ ∈ X \ {α}.

Lemma 5.3.5. Let c : [δ]2 → n be some pair-colouring. Then there existsI ⊆sk δ normal with respect to c.

Proof. By Lemma 5.2.9, it suffices to prove for δ = ωk + 1 for k ∈ N. Wethin out ωk + 1 inductively.

Assume that for every α with CB(α) = m, there exists a function fαsuch that for every β1, β2 ∈ T(α) with β2 <∗ β1, we have c({β1, β2}) =fα(CB(β1),CB(β2)). We replace ωk + 1 with a skeleton J , such that theassumption holds for m+ 1 with respect to c � [J ]2.

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Let α be an ordinal with CB(α) = m+ 1 and enumerate Fan−(α) = {γi :i < ω}. By the pigeonhole principle, we may assume that fγi is constantwith i, and denote it by f . Consider the colouring cα : T(α)→ n and choosesome 0-good Iα ⊆sk T(α) as in Lemma 5.3.2. Let g : m→ n be the functiondefined g(i) = cα(β), where β is any β ∈ Iα with CB(β) = i. So for everyβ1, β2 ∈ I with β2 <

∗ β1 we have

c({β1, β2}) =

{g(CB(β2)), β1 = α

f(CB(β1),CB(β2)), β1 6= α

Replacing T(α) with Iα for each α with CB(α) = m + 1 gives us a skeletonas we seek.

Iterating this process until the stage m = k leaves us with I ⊆sk ωk + 1as described in the statement of the Lemma.

Definition 5.3.6. Let c : [δ]2 → n be some pair-colouring. For α < δ andr ∈ N, if cα is r-good, say that α is r-good for c. Say that α is good for cwhenever α is r-good for c for some natural r.For I ⊆sk δ, α ∈ I, say that α is (r-)good with respect to I for c if ρI(α) is(r-)good for cα ◦ ρ−1

I .

Observation 5.3.7. If J ⊆sk I ⊆sk δ and α ∈ J is r-good with respect to Ifor c, then α is r-good with respect to J for c.

Lemma 5.3.8. Let c : [δ]2 → n be some pair-colouring. Then there existssome I ⊆sk δ such that every α ∈ I is good with respect to I for c.

Proof. Fix some injection σ : δ → ω. We construct inductively {Iα : α < δ}such that

1. I0 = δ

2. Iα ⊆skσ(α)

⋂β<α Iβ;

3. α is good with respect to Iα for c.

By Lemma 5.2.13 and Corollary 5.2.12, I =⋂α<δ Iα will be a skeleton as we

seek.

Let α < δ and assume that Iγ was constructed as above for all γ < α. LetJα =

⋂γ<α Iγ, a skeleton of δ by Lemma 5.2.13 and Corollary 5.2.12. Using

Lemma 5.3.2, let Iα ⊆skσ(α) Jα be σ(α)-good with respect to cα.

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Definition 5.3.9. Let c : [δ]2 → n be some pair-colouring. Let α be goodfor c and denote iα = CNFδ(α).

• For every iα 6= m ≤ kδ, let εm = sup CC δ(m). For each m 6= iα andl < CB(εm), define tα(m, l) to be the unique colour for which

{β ∈ T=l(εm) | c({α, β}) = tα(m, l)} ∈ F(εm)l

Call tα the goodness function of α for c.

Definition 5.3.10. Let c : [δ]2 → n be some pair-colouring. We say that cis uniformly good if

• Every α ∈ I is good with respect to I for c.

• For any α < δ, the goodness function of α depends only on CNFδ(α)and CB(α). Namely, there is a function c, independent of α, such that

tα = c(CNFδ(α),CB(α))

For I ⊆sk δ, say that I is uniformly good with respect to c if c◦ρ−1I is uniformly

good.

Notation 5.3.11. For a function c as in the above definition we abuse no-tation and write

c(i1, j1; i2, j2) := tα(i2, j2)

where α < δ is such that CNFδ(α) = i1 and CB(α) = j1.

Definition 5.3.12. Let c : [δ]2 → n be some pair-colouring. We say thatI ⊆sk δ is canonical with respect to c if I is normal and uniformly good withrespect to c.We say that c is canonical, if δ itself is canonical with respect to c.

Proposition 5.3.13. Let c : [δ]2 → n be some pair-colouring. Then thereexists some I ⊆sk δ canonical with respect to c.

Proof. Let J ⊆ δ be as guaranteed by Lemma 5.3.8.Colour each element of J by its goodness function with respect to J for

c. Exercising Lemma 5.3.2, choose some J ′ such that tα depends only onCNFδ(α) and CB(α). Now use Lemma 5.3.5 on J ′ to receive a skeletonI.

For the purpose of calculating bounds on closed ordinal Ramsey numbers,the above proposition allows us to assume every pair-colouring of δ in finitelymany colours is canonical.

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5.4 Colouring in two colours

For this section fix some ordinal δ < ωω.There is a natural one-to-one correspondence between pair-colourings

c : [δ]2 → 2 and graphs G = (δ, E), given by the rule (α, β) ∈ E ⇐⇒c({α, β}) = 1. We say that a graph on δ is canonical if its correspondingcolouring is canonical.

LetG = (V,E) be a graph. For v ∈ V , denoteN(v) = {u ∈ V | (v, u) ∈ E}.For U ⊆ V , denote N(U) =

⋃v∈U N(v). We say that a set of vertices U ⊆ V

is a clique if [U ]2 ⊆ E and that it is independent if [U ]2 ∩ E = ∅.When the context is clear, we use graph and colouring terminology inter-

changeably. When colouring with colours 0, 1, we will always think of 1 asthe edge colour and of 0 as the non-edge colour.

Definition 5.4.1. Let G = (δ, E) be some graph on δ. Let A,B ⊆ δ beinfinite, disjoint and without maxima.

• Write A ⊥ B to mean that for all X, if X ⊆cof A, then B \ N(X) isfinite.

• Write A ω⊥ B if A ⊥ B and in addition N(a) ∩ B is finite for everya ∈ A.

Note that if A ⊥ B then A0 ⊥ B0 for every A0 ⊆cof A and B0 ⊆ B.

Lemma 5.4.2. Let G = (δ, E). Let A,B ⊆ δ be such that A ω⊥ B. Thenthere is some A0 ⊆cof A and some B0 ⊆ B, cofinite in B, such that N(b)∩A0

is cofinite in A0, for all b ∈ B0.

Proof. We may assume ord(A) = ω by thinning out to a cofinal subset. Wethin out A so either N(a)∩B is constant for all a ∈ A, or N(a)∩B 6= N(b)∩Bfor all distinct a, b ∈ A. Since A ⊥ B, the latter must hold. ConsiderF = {N(a) ∩B | a ∈ A} as a partially ordered set under inclusion. ByRamsey’s theorem, in (F,⊆) there exists either an infinite chain or an infiniteantichain.

Suppose first that there exists some infinite ⊆-antichain C ⊆ F . ByLemma 5.1.1 we choose {ai}i∈ω ⊆ A and {bi}i∈ω ⊆ B such that bi ∈ N(ai) \N({aj}j 6=i). Then for X =

⋃i∈ω a2i, the set B\N(X) is infinite, contradicting

A ⊥ B. Thus, there is some A0 ⊆ A such that {N(a) ∩B | a ∈ A0} is achain in (F,⊆). Let B0 = N(A0) ∩ B, which by A ⊥ B is cofinite in B.

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For every b ∈ B0, since b is an element of the well-ordered ascending union⋃a∈A0

N(a) ∩B, it must be that b ∈ N(a) for cofinitely many a ∈ A0.

5.4.1 Canonical triangle-free graphs

We say that a graph G = (V,E) is triangle-free if N(v) is independent forevery v ∈ V . For the remainder of this section, fix some canonical triangle-free G = (δ, E) with corresponding colouring c : [δ]2 → 2.

Notation 5.4.3. For every i ≤ kδ and j ≤ CB(sup CC δ(i)) denote

Lij = {α ∈ CC δ(i) | CB(α) = j}

Lemma 5.4.4. If there exists no independent X ⊆ δ closed in its supremumwith ord(X) = ω2, then the following statements hold

1. For fixed i, j, there is at most one l such that c(i, j, l) = 1.

2. For fixed i1, i2, j, there is at most one l such that c(i1, j; i2, l) = 1.

3. For fixed i1, i2, l, there is at most one j such that c(i1, j; i2, l) = 1.

Proof. Proof of (1): Assume to the contrary that l1 < l2 are such thatc(i, j, lr) = 1. Let α ∈ Lij. Then T=l1(α) ∪ T=l2(α) ⊆ N(α). Thus, the set

T=l1(α)∪T=l2(α) is independent, and by Corollary 5.2.19 contains a copy ofω2.

Proof of (2): Assume to the contrary that l1 < l2 are such that c(i1, j; i2, lr) =1. Choose arbitrarily some α ∈ Li1j and denote θ = sup CC δ(i2). Let

Fr = N(α) ∩ T=lr(θ) and note Fr ∈ F(γ)lr . Then F1 ∪ F2 is an indepen-dent set, and by Corollary 5.2.19 contains a copy of ω2.

Proof of (3): Assume to the contrary that j1 < j2 are such that c(i1, jr; i2, l) =1. Denote θ = sup CC δ(i1). For any α1, α2 ∈ T=j1(θ) ∪ T=j2(θ), by assump-tion N(α1) ∩ N(α2) ∩ T=l(θ) ∈ F(θ)l. So by triangle-freeness {α1, α2} /∈ E.Thus, T=j1(θ) ∪ T=j2(θ) is an independent set, which contains a copy of ω2

by Corollary 5.2.19.

Lemma 5.4.5. Let B ∈ F(γ)n for some γ ≤ δ. Then there exists someB0 ⊆cof B such that whenever A ⊆ δ is such that A ⊥ B and N(a) ∩ B /∈F(γ)n for each a ∈ A, it is also the case that A ω⊥ B0.

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Proof. For each r ∈ ω, choose arbitrarily some βr ∈ F (γ)rn and let B0 = {βr :r < ω}. Let A be as in the statement. By canonicity, for any α ∈ A we haveB \ N(α) ∈ F(γ)sn for some s and therefore βr /∈ N(α) for all r > s. Thus,N(α) ∩B0 is finite. As B0 ⊆ B, we have that A ω⊥ B0.

Lemma 5.4.6. Let C ⊆ δ. Assume there are some γ < δ, l1 < l2 < CB(γ)and Fi ∈ F(γ)li such that C ⊥ Fi and N(c) ∩ F2 /∈ F(γ)l2 for every c ∈ C.Then for every θ < ω2, there is some independent, closed in its supremum,X ⊆ δ with ord(X) = θ.

Proof. By thinning out C to a cofinal subset, we may assume ord(C) = ω.By Lemma 5.4.5 there is some A2 ⊆cof F2 with C ω⊥ A2. By Lemma 5.4.2,we may assume that N(β)∩C is cofinite in C for each β ∈ A2. Without lossof generality, assume ord(A2) = ω and enumerate A2 = {βi : i < ω}.

Let n ∈ ω be arbitrary, we find in δ an independent copy of ω · (n + 1).The set C ∩

⋂2ni=0 N(βi) is cofinite in C, so without loss of generality equal

to C. For each c ∈ C, let Ic = {i < 2n+ 1 | N(c) ∩ T=l1(βi) ∈ F(βi)l1}. Bythe pigeonhole principle we may assume I := Ic is constant for all c ∈ C.

If |I| > n, choose some α ∈ C, and for each i ∈ I some Xi ⊆cof N(α) ∩T=l1(βi) ∩ F1 with ord(Xi) = ω. Conclude that

⋃i∈I(Xi ∪ {βi}) contains an

independent copy of ω · (n+ 1).Then assume |I| ≤ n, and without loss of generality {0, . . . , n} ∩ I = ∅.

For each i ≤ n, let Bi0 ⊆cof T=l1(βi) ∩ F1 be as guaranteed by Lemma 5.4.5.

As C ⊥ T=l1(βj) ∩ F1 for all j < ω, for i /∈ I we have C ω⊥ Bi0. Iterating

Lemma 5.4.2, construct a descending chain C ⊇ C0 ⊇ C1 · · · ⊇ Cn of infinitesets such that N(α)∩Ci is cofinite in Ci for cofinitely many α ∈ Bi

0. Withoutloss of generality, for every α ∈

⋃i≤nB

i0 the set N(α) ∩ Cn is cofinite in Cn.

Then for any α1, α2 ∈⋃ni=0(Bi

0 ∪ {βi}), the sets N(α1),N(α2) intersect in acofinite subset of Cn. So

⋃ni=0(Bi

0 ∪ {βi}) is an independent set containing acopy of ω · (n+ 1).

Since n was chosen arbitrarily, we find an independent copy of any θ <ω2.

5.5 Rcl(ω · 2, 3) = ω3 · 2For this section, it is helpful to use Figure 5.2. Fix δ = ω3 · 2 and writeLij = {αCC δ(i) | CB(α) = j}.

Proposition 5.5.1. Rcl(ω · 2, 3) ≤ ω3 · 2

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Proof. Assume that the statement is false. Let c : [δ]2 → 2 contradict thestatement and let G = (δ, E) be the triangle-free graph corresponding to c.By Proposition 5.3.13 we may assume that c is canonical.

By item 1 of Lemma 5.4.4, there are j1 < j2 < 3 such that ω3 /∈ N(L1j1∪

L1j2

). By item 2, there are l1 < l2 < 3 such that c(1, 3; 2, l1) = c(1, 3; 2, l2) = 0,and by thinning out to a skeleton we may assume ω3 /∈ N(L2

l1∪ L2

l2).

Assume for a moment that L1jt ⊥ L2

lsfor every t, s ∈ {1, 2}. By Lemma

5.4.6 it is impossible that c(1, jt; 2, l2) = 0. So c(1, j1; 2, l2) = c(1, j2; 2, l2) = 1in contradiction to item 3 of Lemma 5.4.4.

Let, then, t, s ∈ {1, 2} be such that L1jt 6⊥ L2

ls. Let X1 ⊆cof L

1jt be such

that X2 := L2ls\N(X) ⊆cof L

2ls

. By Ramsey’s theorem and triangle-freeness,each Xi contains an infinite independent set. We conclude that X1∪{ω3}∪X2

contains an independent copy of ω · 2 closed in its supremum, contradictingour assumption.

Proposition 5.5.2. Rcl(ω · 2, 3) ≥ ω3 · 2

Proof. We define a (canonical) triangle-free graph G = (δ, E) such that theinduced subgraph on any θ < δ does not contain an independent closed copyof ω ·2. Figure 5.3 is a schematic representation of G. The dashed and dottedlines in the figure imply the ⊥ relation between the circled set more to theleft-hand side and the circled set (or sets) more to the right-hand side.

We use the notation

Ei,jk,l := {(β, α) ∈ E | β < α, β ∈ Lij, α ∈ Lkl }.

We define E so that c(i, j + 1, j) = 1 for all i, j. In our notation,

Ei,ji,j+1 ⊇ {(β, α) ∈ Lij × Lij+1 | β C∗ α}

We list the sets of edges that imply c(i1, j1; i2, j2) = 1:

E1,32,0 = {ω3} × L2

0

E1,21,0 = {(ω2 · k, ω2 · k′ + ω · l′ +m′) | 0 < k ≤ k′}

E1,01,2 = {(ω2 · k + ω · l +m, ω2 · k′) | (k + 1) + l < k′}

E1,02,1 = {(ω2 · k + ω · l +m, ω3 + ω2 · k′ + ω · l′) | k + l < l′}

E2,22,0 = {(ω3 + ω2 · k, ω3 + ω2 · k′ + ω · l′ +m′) | k ≤ k′}

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Lastly, we list sets of edges that imply ⊥ while c(i1, j1; i2, j2) = 0:

E1,12,1 ={(ω2 · k + ω · l, ω3 + ω2 · k′ + ω · l′) | l′ < k}

E1,01,1 ⊇{(ω2 · k + ω · l +m, ω2 · k′ + ω · l′) | 0 < k′ − k < l, l′ < l}

E1,02,0 ={(ω2 · k + ω · l +m, ω3 + ω2 · k′ + ω · l′ +m′) | l′ < l}

E2,02,1 ⊇{(ω3 + ω2 · k + ω · l +m, ω3 + ω2 · k′ + ω · l′) | k < k′, l′ < l}

In the last list, for E1,01,1 and E2,0

2,1 we do not use the equality sign in orderto accommodate edges whose existence was declared previously. We let E beminimal under inclusion and fulfilling our stated requirements.

Claim 1. G = (δ, E) is triangle-free

Proof. Note that Lij is independent for all i, j.Assume {ω3, α, β} is a triangle with α < β. Then it must be that α ∈ L1

2

and β ∈ L20. But then, it cannot be that {α, β} ∈ E. So ω3 takes no part in

a triangle.Consider γ ∈ L1

2. Assume T = {α, β, γ} is a triangle with α < β.Looking at N(L1

2), it must be that T ⊆ CC δ(1). Since ω3 /∈ T it must bethat {CB(α),CB(β)} = {0, 1}. Since E1,1

1,0 = ∅, we have CB(α) = 0 and

CB(β) = 1. Since E1,21,1 = ∅ we have that (β, γ) ∈ E1,1

1,2 , implying β <∗ γ and

α < β < γ. From (α, γ) ∈ E1,01,2 we get α 6<∗ γ and so α 6<∗ β. So

α = ω2 · k + ω · l +m

β = ω2 · (k′ − 1) + ω · l′

γ = ω2 · k′

with the restrictions (k′−1)−k < l and (k+1)+ l < k′ coming from (α, β) ∈E1,0

1,1 with α 6<∗ β and (α, γ) ∈ E1,01,2 respectively. These two inequalities are

contradictory, so γ cannot take part in a triangle.Consider β ∈ L1

1. Assume {α, β, γ} is a triangle with α < γ. We alreadyhave enough to guarantee that there are no triangles within CC δ(1). Soγ ∈ CC δ(2) and in particular (β, γ) ∈ E1,1

2,1 . This leaves no other option but

(α, β) ∈ E1,01,1 . So

α = ω2 · k + ω · l +m

β = ω2 · k′ + ω · l′

γ = ω3 + ω2 · k′′ + ω · l′′

87

with the restrictions k + l < l′′ and l′′ < k′ coming from (α, γ) ∈ E1,02,1 and

(β, γ) ∈ E1,12,1 respectively, which gives k+ l < k′. As k′ 6= k, it cannot be that

αC∗β. Thus, we have the additional restriction k′−k < l from (α, β) ∈ E1,01,1 .

This again is contradictory, and so β takes no part in a triangle.Consider β ∈ L2

2. Assume {α, β, γ} is a triangle with α < γ. It must bethat α < β < γ, αC∗ β and γ ∈ L2

0. But E2,12,0 = ∅, and so (α, γ) ∈ E cannot

happen, contrary to our assumption.Finally, assume {α, β, γ} is a triangle with α < β < γ. The only possi-

bility left is α ∈ L10, β ∈ L2

0, γ ∈ L21. So

α = ω2 · k + ω · l +m

β = ω3 + ω2 · k′ + ω · l′ +m′

γ = ω3 + ω2 · k′′ + ω · l′′

with l′ < l coming from (α, β) ∈ E1,02,0 and k+l < l′′ coming from (α, γ) ∈ E1,0

2,1 .If βC∗γ, then l′′ = l′+1, leading to k+l < l′+1 < l+1 which is impossible. Sothis is not the case, and we get the extra restriction l′′ < l′ from (β, γ) ∈ E2,0

2,1 .This gives us k + l < l′ < l in contradiction. All possibilities have beenexhausted, and so there are no triangles in G.

Claim 2. Every independent copy of ω · 2 contained in G, which is closed inits supremum, is cofinal in δ.

Proof. Let W := A ∪ {τ} ∪ B ⊆ δ be an independent set with A < τ < B,ord(A) = ord(B) = ω and τ = supA. Assume for the purpose of a contradic-tion that W ⊆ θ for some θ < δ. Without loss of generality, let θ = ω3 +ω2 ·nfor some natural n. For i, j denote L(θ)ij = {α ∈ CC θ(i) | CB(α) = j}.

By the pigeonhole principle, we may assume that A ⊆cof T=jA(τ) for somejA < CB(τ), and B ⊆ L(θ)iBlB for some iB ≤ kθ, lB < CB(sup CC θ(iB)). Fixsuch jA, iB, lB. We proceed by eliminating where τ can potentially reside.

As τ is a limit ordinal, CB(t) > 0. Since Fan−(τ) × t ⊆ E, it also mustbe that CB(τ) > 1 and jA < CB(τ)− 1.

Assume τ ∈ L(θ)13, that is τ = ω3. Then clearly iB > 1, lB = 1. For every

α ∈ L(θ)10 the set L(θ)iB1 \ N(α) is finite, so jA 6= 0. Also, L(θ)1

1 ω⊥ L(θ)iB1 ,and so jA 6= 1. This contradicts jA < CB(t)−1 = 2, and so τ /∈ L(θ)1

3. Thus,it must be that CB(τ) = 2 and jA = 0.

Assume τ ∈ L12. Note the following:

88

• {α > τ} ∩ L(θ)10 ⊆ N(τ);

• {α > τ} ∩ L(θ)11 ⊆ N(A);

• L(θ)12 \N(α) is finite for every α ∈ A.

The three listed items guarantee that iB > 1. However, for each i > 1 notethat

• L(θ)i0 ⊆ N(A);

• L(θ)i1 \N(α) is finite for every α ∈ A.

So it cannot be that lB ∈ {0, 1}, a contradiction.So the only option is that τ ∈ L(θ)i2 for some 1 < i ≤ kθ. We now observe

that for each i′ > i

• L(θ)i′

0 ⊆ N(τ);

• L(θ)i′

1 ⊆ N(A).

prohibiting lB = 0 or lB = 1, and therefore the existence of lB. We concludethat there is no θ as we had assumed.

By the claims, taking the induced colouring on any θ < ω3·2 demonstratesthat θ 6→cl (ω · 2, 3)2.

Corollary 5.5.3. Rcl(ω · 2, 3) = ω3 · 2

89

Fig

ure

5.2:ω

3·2

under

the<∗

rela

tion

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

ω3

ω2

ω2·2

ω2·3

ωω·2

ω·3

01

2

ω3·2

ω3

+ω

2ω

3+ω

2·2ω

3+ω

2·3

ω3

+ω

ω3+

1

90

Fig

ure

5.3:

Vis

ual

izat

ion

ofL

emm

a5.

5.2

91

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