Applications of High Dimensional Ellentuck spaces

Natasha Dobrinen

University of Denver

Spring Topology and Dynamics Conference, 2018

Research supported by NSF Grant DMS-1600781

Dobrinen High dimensional Ellentuck University of Denver 1 / 21

BLAST Conference - 10 Year Anniversary

Boolean algebrasLatticesAlegbraic logic, universal AlgebraSet theoryTopology - general, point-free, set-theoretic

University of DenverAugust 6 - 10, 2018

http://www.cs.du.edu/∼wesfussn/blastsome travel/lodging support available

Dobrinen High dimensional Ellentuck University of Denver 2 / 21

Ellentuck Space

The Ellentuck space is the space [ω]ω with topology generated by basicopen sets

[s,A] = {X ∈ [ω]ω : s < X ⊆ A},

where s ∈ [ω]<ω and A ∈ [ω]ω.

Thm. (Ellentuck) The Ellentuck space is a topological Ramsey space:Given X ⊆ [ω]ω with the property of Baire, for any basic open set[s,A], there is a member B ∈ [s,A] such that

either [s,B] ⊆ X or else [s,B] ∩ X = ∅.

Forcing with members of the Ellentuck space partially ordered by ⊆∗adds a Ramsey ultrafilter.

Dobrinen High dimensional Ellentuck University of Denver 3 / 21

Forcing Ultrafilters

([ω]ω,⊆∗) is forcing equivalent to P(ω)/Fin.

A natural extension of this Boolean algebra: P(ω × ω)/Fin⊗ Fin.

X ∈ Fin⊗ Fin iff X ⊆ ω × ω and ∀∞n ∈ ω, {i ∈ ω : (n, i) ∈ X} ∈ Fin.

P(ω2)/Fin⊗2 adds an ultrafilter U2, the next best thing to a p-point:

U2 → (U2)2r ,4.

The projection to the first coordinates, π1(U2), is a Ramsey ultrafilter,generic for π1(P(ω2)/Fin⊗2) ∼= P(ω)/Fin.

Dobrinen High dimensional Ellentuck University of Denver 4 / 21

Extending Fin⊗2 to all uniform barriers

Recursively construct ideals on ωk+1: Fin⊗k+1 = Fin⊗ Fin⊗k .

P(ωk)/Fin⊗k forces an ultrafilter Uk : for each j < k , πj(Uk) ∼= Uj .

Replace ωk by [ω]k ; Fin⊗k by the ideal Ik on [ω]k determined by Fin⊗k .

P([ω]k)/Ik ∼= P(ωk)/Fin⊗k .

[ω]k is a uniform barrier on ω of rank k.

This construction of Ik can be extended to all uniform barriers on ω.

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Hierarchy of Boolean Algebras and forced ultrafilters

Example. Schreier barrier: S = {s ∈ [ω]<ω : |s| = min s + 1}.

For X ⊆ S , Xn = {s ∈ X : min s = n}.

IS = {X ⊆ S : ∀∞n (Xn ∈ ISn)}.

For any uniform barrier B on ω, P(B)/IB forces an ultrafilter UB oncountable base set B.

Fact. If B projects to C , then P(C )/IC embeds as a completesubalgebra of P(B)/IB , and UC is isomorphic to a projection of UB .

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Initial motivation for h.d. Ellentuck spaces: Cofinal Types

A function f : U → V between ultrafilters is cofinal if f maps each filterbase for U to a filter base for V.

U is Tukey reducible to U ≥T V iff there is a cofinal map from U into V.

The equivalence relation defined by U ≡T V iff U ≤T V and V ≤T U is acoarsening of the Rudin-Keisler equivalence relation of isomorphism.

Thm.

1 (Folklore) The ultrafilter U2 forced by P(ω2)/Fin⊗2 isRudin-Keisler minimal above the Ramsey ultrafilter π1(U2).

2 (Blass, D., Raghavan) U2 ≥T π1(U2) and U2 is not Tukey maximal.

Dobrinen High dimensional Ellentuck University of Denver 7 / 21

Initial Tukey Structures

So what exactly is Tukey below U2?

Thm. [D1]

1 U2 is Tukey minimal above its projected Ramsey ultrafilter π1(U2).

2 For each k ≥ 2, The ultrafilter Uk forced by P(ωk)/Fin⊗k hasinitial Tukey structure exactly a chain of length k . Likewise for itsinitial Rudin-Keisler structure.

3 [D2 and unpublished] For each uniform barrier B of infinite rank,UB has initial Tukey and RK structures which are chains of length2ω, and they form a hierarchy via projection to barriers of smallerrank.

Remark. These results rely on new topological Ramsey spaces andcanonization theorems for equivalence relations.

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The 2-dimensional Ellentuck space E2

Goal: Construct a topological Ramsey space dense in (Fin⊗ Fin)+.

Q. Which subsets of (Fin⊗ Fin)+ should we allow?

A. Fix a particular order ≺ of the members of non-decreasingsequences of natural numbers of length 2 in order type ω so that eachinfinite set is the limit of its finite approximations.

∅

(5)

(5, 5)

(4)

(4, 5)(4, 4)

(3)

(3, 5)(3, 4)(3, 3)

(2)

(2, 5)(2, 4)(2, 3)(2, 2)

(1)

(1, 5)(1, 4)(1, 3)(1, 2)(1, 1)

(0)

(0, 5)(0, 4)(0, 3)(0, 2)(0, 1)(0, 0)

Figure: W2

E2 consists of all subsets of W2 for which the ≺-preserving bijection isalso a tree-isomorphism.

Dobrinen High dimensional Ellentuck University of Denver 9 / 21

〈〉

0 1 2 3 4 5

001002030405 1112 22232425 3334 4445 555657

A member of E2

〈〉

0 1 2 3 4 5

00 10 11 12 13 14 15 22 23 33 34 35 36 44 45 55 56

6

66 67

A member of E2

Dobrinen High dimensional Ellentuck University of Denver 10 / 21

The collection of X ⊆W2 for which the ≺-preserving bijection from W2

to X preserves the tree structure induces the finite approximations.The basic open sets of E2 are of the form

[s,A] = {X ∈ E2 : s < A ⊆ X}.

Thm. [D1] E2 satisfies the 4 axioms of Todorcevic, and hence is atopological Ramsey space: Every subset with the property of Baire isRamsey.

That E2 is a topological Ramsey space was heavily utilized whenproving the canonization theorem for equivalence relations on barrierson E2.

This was applied to show that the generic ultrafilter forced byP(ω2)/Fin⊗2 has, up to cofinal equivalence, exactly one Tukey typebelow it, namely that of its projected Ramsey ultrafilter.

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The 3-dimensional Ellentuck space E3

∅

(3)

(3, 3)

(3,3,3)

(2)

(2, 3)

(2,3,3)

(2, 2)

(2,2,3)

(2,2,2)

(1)

(1, 3)

(1,3,3)

(1, 2)

(1,2,3)

(1,2,2)

(1, 1)

(1,1,3)

(1,1,2)

(1,1,1)

(0)

(0, 3)

(0,3,3)

(0, 2)

(0,2,3)

(0,2,2)

(0, 1)

(0,1,3)

(0,1,2)

(0,1,1)

(0, 0)

(0,0,3)

(0,0,2)

(0,0,1)

(0,0,0)

Figure: W3

〈〉0 1 2 3 4 5

A member of E3Dobrinen High dimensional Ellentuck University of Denver 12 / 21

ES for S the Schreier barrier

∅

(3)

(3, 4)

(3, 4, 4)

(3,4,4,4)

(3, 3)

(3, 3, 4)

(3,3,4,4)

(3, 3, 3)

(3,3,3,4)

(3,3,3,3)

(2)

(2, 4)

(2,4,4)

(2, 3)

(2,3,4)

(2,3,3)

(2, 2)

(2,2,4)

(2,2,3)

(2,2,2)

(1)

(1, 4)(1, 3)(1, 2)(1, 1)

(0)

Figure: WS

X ∈ ES only if X ⊆WS , for each n for which X has non-emptyintersection with the subtree above (n), that restriction of X is in En, andmore structural requirements which are defined recursively from thestructural requirements for th Ek .

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The Infinite Dimensional Ellentuck Spaces

Thm. [D2] For each uniform barrier B, there is a topological Ramseyspace EB which is dense in I+B .

Hence, (EB ,⊆IB ) is forcing equivalent to P(B)/IB .

Thus, the restriction of P(B) to EB produces infinitary Ramsey theory,for those partitions into sets satisfying the property of Baire in theEllentuck topology.

This was a necessary, though not sufficient, step in proving the initialTukey structures below the ultrafilters UB .

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Other Applications of Extended Ellentuck Spaces

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A hierarchy of new Banach spaces

In [Arias, D., Giron, Mijares], we constructed a new Banach spaces usingthe Tsirelson norm construction over fronts of finite rank on the Ek spaces.

This forms a hierarchy of spaces over `p, with spaces formed from Ekprojecting (in many different ways) to spaces from Ej , for j < k .

My motivation for this project was to shed new light on distortionproblems. Much work still needs to be done in this direction.

Dobrinen High dimensional Ellentuck University of Denver 16 / 21

Preservation of ultrafilters by Product Sacks Forcing

Thm. [Y.Y. Zheng] The ultrafilters forced by P(ωk)/Fin⊗k arepreserved by products of Sacks forcing with countable support.

She first proved a Moderately-Abstract Parametrized EllentuckTheorem for R× Rω, for a large class of topological Ramsey spaces.

She then showed that Ek spaces satisfy the premises of thisparametrization theorem, which is applied to obtain the theorem above.

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A Barren Extension

Thm. [Henle, Mathias, Woodin] Let M be a transitive model of ZF +ω → (ω)ω and N its Hausdorff extension, that is the extension M[U ]where U is the Ramsey ultrafilter forced by P(ω)/Fin.

Then M and N have the same sets of ordinals; moreover, everysequence in N of elements of M lies in M.

In particular, this theorem holds when M is the Solovay model L(R).

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A Hierarchy of Barren Extensions

Thm. [D., Hathaway] Fix a uniform barrier B. Let M be a transitivemodel of ZF + every subset of EB is Ramsey, and let N = M[UB ] be thegeneric extension obtained by forcing with (EB ,⊆IB ).

Then M and N have the same sets of ordinals; moreover, everysequence in N of elements of M lies in M.

Thus, there is a hierarchy of models L(R)[UB ] with stronger andstronger fragments of choice, in the form of containing an ultrafilter UBand all UC where C is a uniform barrier obtained by a projection of B,all of which are barren extenions of L(R).

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References

[Arias, D., Giron, Mijares] Banach spaces from barriers in high dimensionalEllentuck spaces (2017) (preprint) 25pp.

[Blass, D., Raghavan] The next best thing to a p-point, JSL (2015) 44pp.

[D1] High dimensional Ellentuck spaces and initial chains in the Tukeystructure of non-p-points, JSL (2016) 26pp.

[D2] Infinite dimensional Ellentuck spaces and Ramsey-classificationtheorems JML (2016) 37pp.

[D., Hathaway] A family of barren extensions, In preparation.

[Henle, Mathias, Woodin] A barren extension, Lecture Notes in Math.,1130, Springer (1985) 12pp.

[Zheng] Parametrizing topological Ramsey spaces, 27 pp. Submitted.

[Todorcevic] Introduction to Ramsey Spaces Princeton University Press(2010).

Dobrinen High dimensional Ellentuck University of Denver 20 / 21

BLAST Conference - 10 Year Anniversary

Boolean algebrasLatticesAlegbraic logic, universal AlgebraSet theoryTopology - general, point-free, set-theoretic

University of DenverAugust 6 - 10, 2018

http://www.cs.du.edu/∼wesfussn/blastsome travel/lodging support available thanks to NSF funding

Come have a BLAST!

Dobrinen High dimensional Ellentuck University of Denver 21 / 21