Higher dimensional Ellentuck spaces Natasha Dobrinen University of - - PowerPoint PPT Presentation

Higher dimensional Ellentuck spaces

Natasha Dobrinen University of Denver Forcing and Its Applications Retrospective Workshop Fields Institute, April 1, 2015

Dobrinen Higher dimensional Ellentuck spaces University of Denver 1 / 43

References for this talk

This work was initially motivated by work done at the Fields in 2012 in [Blass/Dobrinen/Raghavan] The next best thing to a p-point, submitted.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 2 / 43

References for this talk

This work was initially motivated by work done at the Fields in 2012 in [Blass/Dobrinen/Raghavan] The next best thing to a p-point, submitted. Work in this talk

Dobrinen Higher dimensional Ellentuck spaces University of Denver 2 / 43

References for this talk

This work was initially motivated by work done at the Fields in 2012 in [Blass/Dobrinen/Raghavan] The next best thing to a p-point, submitted. Work in this talk Ek (2 ≤ k < ω) spaces appear in [Dobrinen] High dimensional Ellentuck spaces and initial chains in the Tukey types of non p-points, Journal of Symbolic Logic, to appear.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 2 / 43

References for this talk

This work was initially motivated by work done at the Fields in 2012 in [Blass/Dobrinen/Raghavan] The next best thing to a p-point, submitted. Work in this talk Ek (2 ≤ k < ω) spaces appear in [Dobrinen] High dimensional Ellentuck spaces and initial chains in the Tukey types of non p-points, Journal of Symbolic Logic, to appear. EB spaces are part of current investigation.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 2 / 43

References for this talk

This work was initially motivated by work done at the Fields in 2012 in [Blass/Dobrinen/Raghavan] The next best thing to a p-point, submitted. Work in this talk Ek (2 ≤ k < ω) spaces appear in [Dobrinen] High dimensional Ellentuck spaces and initial chains in the Tukey types of non p-points, Journal of Symbolic Logic, to appear. EB spaces are part of current investigation. For a survey of pre-2014 Tukey and related Ramsey space results, see [Dobrinen] Survey on the Tukey theory of ultrafilters, Zbornik Radova, Mathematical Institute of the Serbian Academy of Sciences, 2015.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 2 / 43

A very brief review of Tukey reduction between ultrafilters

• Def. V is Tukey reducible to U (V ≤T U) if there is a map f : U → V

such that each f -image of a filter base for U is a filter base for V.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 3 / 43

A very brief review of Tukey reduction between ultrafilters

• Def. V is Tukey reducible to U (V ≤T U) if there is a map f : U → V

such that each f -image of a filter base for U is a filter base for V. U ≡T V iff U ≤T V and V ≤T U. The Tukey equivalence class of an ultrafilter is called its Tukey type.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 3 / 43

A very brief review of Tukey reduction between ultrafilters

• Def. V is Tukey reducible to U (V ≤T U) if there is a map f : U → V

such that each f -image of a filter base for U is a filter base for V. U ≡T V iff U ≤T V and V ≤T U. The Tukey equivalence class of an ultrafilter is called its Tukey type. V ≤RK U iff ∃f : ω → ω such that {f (U) : U ∈ U} generates V. For ultrafilters, Rudin-Keisler reduction implies Tukey reduction. Thus, Tukey types are a coarsening of Rudin-Keisler (isomorphism) equivalence classes of ultrafilters.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 3 / 43

A very brief review of Tukey reduction between ultrafilters

• Def. V is Tukey reducible to U (V ≤T U) if there is a map f : U → V

such that each f -image of a filter base for U is a filter base for V. U ≡T V iff U ≤T V and V ≤T U. The Tukey equivalence class of an ultrafilter is called its Tukey type. V ≤RK U iff ∃f : ω → ω such that {f (U) : U ∈ U} generates V. For ultrafilters, Rudin-Keisler reduction implies Tukey reduction. Thus, Tukey types are a coarsening of Rudin-Keisler (isomorphism) equivalence classes of ultrafilters. For more overview, see my recent survey paper.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 3 / 43

Prior to work in this talk, quite a bit had been done finding embedded structures and initial structures of Tukey (and RK) types of p-points and iterated Fubini products of p-points.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 4 / 43

Prior to work in this talk, quite a bit had been done finding embedded structures and initial structures of Tukey (and RK) types of p-points and iterated Fubini products of p-points. [Milovich 2008 (initial work on Tukey and Isbell’s Problem)] [Dobrinen/Todorcevic 2011 (embeddings), 2014 and 2015 (initial structures)] [Dobrinen Continuous cofinal maps 2010 preprint - (extended to become Continuous and other canonical cofinal maps (2015)) ] [Raghavan/Todorcevic 2012 (RK versus Tukey and first initial structure result for Ramsey ultrafilters)] [Dobrinen/Mijares/Trujillo submitted 2014 (Boolean algebras as initial structures for Tukey and a rich collection of initial structures for RK)] [Raghavan/Shelah submitted 2014 (embedding P(ω)/fin into RK and Tukey types of p-points)]

Dobrinen Higher dimensional Ellentuck spaces University of Denver 4 / 43

The Forcing P(ω × ω)/Fin⊗2

Fin ⊗ Fin = {X ⊆ ω × ω : ∀∞i ∈ ω {j ∈ ω : (i, j) ∈ X} is finite}. That is, for all but finitely many i, the i-th fiber of X is finite.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 5 / 43

The Forcing P(ω × ω)/Fin⊗2

Fin ⊗ Fin = {X ⊆ ω × ω : ∀∞i ∈ ω {j ∈ ω : (i, j) ∈ X} is finite}. That is, for all but finitely many i, the i-th fiber of X is finite. We also use Fin⊗2 to denote Fin ⊗ Fin.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 5 / 43

The Forcing P(ω × ω)/Fin⊗2

Fin ⊗ Fin = {X ⊆ ω × ω : ∀∞i ∈ ω {j ∈ ω : (i, j) ∈ X} is finite}. That is, for all but finitely many i, the i-th fiber of X is finite. We also use Fin⊗2 to denote Fin ⊗ Fin. P(ω × ω)/Fin⊗2 forces a generic ultrafilter G2 on base set ω × ω.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 5 / 43

The Forcing P(ω × ω)/Fin⊗2

Fin ⊗ Fin = {X ⊆ ω × ω : ∀∞i ∈ ω {j ∈ ω : (i, j) ∈ X} is finite}. That is, for all but finitely many i, the i-th fiber of X is finite. We also use Fin⊗2 to denote Fin ⊗ Fin. P(ω × ω)/Fin⊗2 forces a generic ultrafilter G2 on base set ω × ω. G2 is neither a p-point, nor a Fubini product of p-points, but the projection to the first coordinates π1(G2) is a Ramsey ultrafilter.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 5 / 43

In [Blass/Dobrinen/Raghavan], we showed the following:

• Thm. In V [G2], G2

(B) is a weak p-point;

Dobrinen Higher dimensional Ellentuck spaces University of Denver 6 / 43

In [Blass/Dobrinen/Raghavan], we showed the following:

• Thm. In V [G2], G2

(B) is a weak p-point; (B) has the best partition property G2 → (G2)2

k,3 a non-p-point can

have;

Dobrinen Higher dimensional Ellentuck spaces University of Denver 6 / 43

In [Blass/Dobrinen/Raghavan], we showed the following:

• Thm. In V [G2], G2

(B) is a weak p-point; (B) has the best partition property G2 → (G2)2

k,3 a non-p-point can

have; (D,R) is not Tukey maximum;

Dobrinen Higher dimensional Ellentuck spaces University of Denver 6 / 43

In [Blass/Dobrinen/Raghavan], we showed the following:

• Thm. In V [G2], G2

(B) is a weak p-point; (B) has the best partition property G2 → (G2)2

k,3 a non-p-point can

have; (D,R) is not Tukey maximum; (D) (G2, ⊇) ≥T ([ω1]<ω, ⊆);

Dobrinen Higher dimensional Ellentuck spaces University of Denver 6 / 43

In [Blass/Dobrinen/Raghavan], we showed the following:

• Thm. In V [G2], G2

(B) is a weak p-point; (B) has the best partition property G2 → (G2)2

k,3 a non-p-point can

have; (D,R) is not Tukey maximum; (D) (G2, ⊇) ≥T ([ω1]<ω, ⊆); (D) G2 >T π1(G2);

Dobrinen Higher dimensional Ellentuck spaces University of Denver 6 / 43

In [Blass/Dobrinen/Raghavan], we showed the following:

• Thm. In V [G2], G2

(B) is a weak p-point; (B) has the best partition property G2 → (G2)2

k,3 a non-p-point can

have; (D,R) is not Tukey maximum; (D) (G2, ⊇) ≥T ([ω1]<ω, ⊆); (D) G2 >T π1(G2); (R) is not ‘basically generated’.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 6 / 43

In [Blass/Dobrinen/Raghavan], we showed the following:

• Thm. In V [G2], G2

(B) is a weak p-point; (B) has the best partition property G2 → (G2)2

k,3 a non-p-point can

have; (D,R) is not Tukey maximum; (D) (G2, ⊇) ≥T ([ω1]<ω, ⊆); (D) G2 >T π1(G2); (R) is not ‘basically generated’. This left open what exactly is Tukey reducible to G2; i.e. What is the initial Tukey structure below G2.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 6 / 43

P(ω × ω)/Fin⊗2 is forcing equivalent to ((Fin ⊗ Fin)+, ⊆Fin

⊗2

),

Dobrinen Higher dimensional Ellentuck spaces University of Denver 7 / 43

P(ω × ω)/Fin⊗2 is forcing equivalent to ((Fin ⊗ Fin)+, ⊆Fin

⊗2

), which is forcing equivalent to {X ⊆ ω × ω : infinitely many fibers of X are infinite, and all finite fibers of X are empty}, partially ordered by ⊆Fin

⊗2

.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 7 / 43

P(ω × ω)/Fin⊗2 is forcing equivalent to ((Fin ⊗ Fin)+, ⊆Fin

⊗2

), which is forcing equivalent to {X ⊆ ω × ω : infinitely many fibers of X are infinite, and all finite fibers of X are empty}, partially ordered by ⊆Fin

⊗2

. We will thin this even more and put more restrictions on the subsets of ω × ω we allow in order to obtain a topological Ramsey space E2 which is forcing equivalent to P(ω × ω)/Fin⊗2. Our space E2 looks and acts like ω copies of the Ellentuck space, given a judiciously chosen finitization map.

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Review

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Simplest Topological Ramsey Space: The Ellentuck Space

• Example. Ellentuck space [ω]ω. Y ≤ X iff Y ⊆ X.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 9 / 43

Simplest Topological Ramsey Space: The Ellentuck Space

• Example. Ellentuck space [ω]ω. Y ≤ X iff Y ⊆ X.

Basis for topology: [s, X] = {Y ∈ [ω]ω : s ❁ Y ⊆ X}.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 9 / 43

Simplest Topological Ramsey Space: The Ellentuck Space

• Example. Ellentuck space [ω]ω. Y ≤ X iff Y ⊆ X.

Basis for topology: [s, X] = {Y ∈ [ω]ω : s ❁ Y ⊆ X}.

• Def. X ⊆ [ω]ω is Ramsey iff for each [s, X], there is s ❁ Y ⊆ X such

that either [s, Y ] ⊆ X or [s, Y ] ∩ X = ∅.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 9 / 43

Simplest Topological Ramsey Space: The Ellentuck Space

• Example. Ellentuck space [ω]ω. Y ≤ X iff Y ⊆ X.

Basis for topology: [s, X] = {Y ∈ [ω]ω : s ❁ Y ⊆ X}.

• Def. X ⊆ [ω]ω is Ramsey iff for each [s, X], there is s ❁ Y ⊆ X such

that either [s, Y ] ⊆ X or [s, Y ] ∩ X = ∅.

• Thm. [Ellentuck 1974] Every X ⊆ [ω]ω with the property of Baire (in

the Ellentuck topology) is Ramsey.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 9 / 43

Simplest Topological Ramsey Space: The Ellentuck Space

• Example. Ellentuck space [ω]ω. Y ≤ X iff Y ⊆ X.

Basis for topology: [s, X] = {Y ∈ [ω]ω : s ❁ Y ⊆ X}.

• Def. X ⊆ [ω]ω is Ramsey iff for each [s, X], there is s ❁ Y ⊆ X such

that either [s, Y ] ⊆ X or [s, Y ] ∩ X = ∅.

• Thm. [Ellentuck 1974] Every X ⊆ [ω]ω with the property of Baire (in

the Ellentuck topology) is Ramsey. Galvin-Prikry Theorem: All (metrically) Borel sets are Ramsey.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 9 / 43

Simplest Topological Ramsey Space: The Ellentuck Space

• Example. Ellentuck space [ω]ω. Y ≤ X iff Y ⊆ X.

Basis for topology: [s, X] = {Y ∈ [ω]ω : s ❁ Y ⊆ X}.

• Def. X ⊆ [ω]ω is Ramsey iff for each [s, X], there is s ❁ Y ⊆ X such

that either [s, Y ] ⊆ X or [s, Y ] ∩ X = ∅.

• Thm. [Ellentuck 1974] Every X ⊆ [ω]ω with the property of Baire (in

the Ellentuck topology) is Ramsey. Galvin-Prikry Theorem: All (metrically) Borel sets are Ramsey. Silver Theorem: All (metrically) Suslin sets are Ramsey.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 9 / 43

Simplest Topological Ramsey Space: The Ellentuck Space

• Example. Ellentuck space [ω]ω. Y ≤ X iff Y ⊆ X.

Basis for topology: [s, X] = {Y ∈ [ω]ω : s ❁ Y ⊆ X}.

• Def. X ⊆ [ω]ω is Ramsey iff for each [s, X], there is s ❁ Y ⊆ X such

that either [s, Y ] ⊆ X or [s, Y ] ∩ X = ∅.

• Thm. [Ellentuck 1974] Every X ⊆ [ω]ω with the property of Baire (in

the Ellentuck topology) is Ramsey. Galvin-Prikry Theorem: All (metrically) Borel sets are Ramsey. Silver Theorem: All (metrically) Suslin sets are Ramsey. Associated Forcings: Mathias, P(ω)/fin.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 9 / 43

Simplest Topological Ramsey Space: The Ellentuck Space

• Example. Ellentuck space [ω]ω. Y ≤ X iff Y ⊆ X.

Basis for topology: [s, X] = {Y ∈ [ω]ω : s ❁ Y ⊆ X}.

• Def. X ⊆ [ω]ω is Ramsey iff for each [s, X], there is s ❁ Y ⊆ X such

that either [s, Y ] ⊆ X or [s, Y ] ∩ X = ∅.

• Thm. [Ellentuck 1974] Every X ⊆ [ω]ω with the property of Baire (in

the Ellentuck topology) is Ramsey. Galvin-Prikry Theorem: All (metrically) Borel sets are Ramsey. Silver Theorem: All (metrically) Suslin sets are Ramsey. Associated Forcings: Mathias, P(ω)/fin. Associated Ultrafilter: Ramsey ultrafilter forced by ([ω]ω, ≤∗), has ‘complete combinatorics’.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 9 / 43

Topological Ramsey spaces (R, ≤, r)

Basic open sets: [a, A] = {X ∈ R : ∃n(rn(X) = a) and X ≤ A}.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 10 / 43

Topological Ramsey spaces (R, ≤, r)

Basic open sets: [a, A] = {X ∈ R : ∃n(rn(X) = a) and X ≤ A}.

• Def. X ⊆ R is Ramsey iff for each ∅ = [a, A], there is a B ∈ [a, A] such

that either [a, B] ⊆ X or [a, B] ∩ X = ∅.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 10 / 43

Topological Ramsey spaces (R, ≤, r)

Basic open sets: [a, A] = {X ∈ R : ∃n(rn(X) = a) and X ≤ A}.

• Def. X ⊆ R is Ramsey iff for each ∅ = [a, A], there is a B ∈ [a, A] such

that either [a, B] ⊆ X or [a, B] ∩ X = ∅.

• Def. [Todorcevic] A triple (R, ≤, r) is a topological Ramsey space if

every subset of R with the Baire property is Ramsey, and if every meager subset of R is Ramsey null.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 10 / 43

Topological Ramsey spaces (R, ≤, r)

Basic open sets: [a, A] = {X ∈ R : ∃n(rn(X) = a) and X ≤ A}.

• Def. X ⊆ R is Ramsey iff for each ∅ = [a, A], there is a B ∈ [a, A] such

that either [a, B] ⊆ X or [a, B] ∩ X = ∅.

• Def. [Todorcevic] A triple (R, ≤, r) is a topological Ramsey space if

every subset of R with the Baire property is Ramsey, and if every meager subset of R is Ramsey null. Abstract Ellentuck Theorem. [Todorcevic] If (R, ≤, r) satisfies A.1 - A.4 and R is closed (in ARN), then (R, ≤, r) is a topological Ramsey space.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 10 / 43

Topological Ramsey spaces (R, ≤, r)

Basic open sets: [a, A] = {X ∈ R : ∃n(rn(X) = a) and X ≤ A}.

• Def. X ⊆ R is Ramsey iff for each ∅ = [a, A], there is a B ∈ [a, A] such

that either [a, B] ⊆ X or [a, B] ∩ X = ∅.

• Def. [Todorcevic] A triple (R, ≤, r) is a topological Ramsey space if

every subset of R with the Baire property is Ramsey, and if every meager subset of R is Ramsey null. Abstract Ellentuck Theorem. [Todorcevic] If (R, ≤, r) satisfies A.1 - A.4 and R is closed (in ARN), then (R, ≤, r) is a topological Ramsey space. n-th Appproximations: ARn = {rn(X) : X ∈ R}. Finite Approximations: AR =

n<ω ARn.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 10 / 43

tRs’s force ultrafilters with complete combinatorics

• Thm. [DiPrisco/Mijares/Nieto (submitted 2014)] Let R be a

topological Ramsey space. If there exists a supercompact cardinal, then every selective coideal U ⊆ R is (R, ≤∗)-generic over L(R).

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tRs’s force ultrafilters with complete combinatorics

• Thm. [DiPrisco/Mijares/Nieto (submitted 2014)] Let R be a

topological Ramsey space. If there exists a supercompact cardinal, then every selective coideal U ⊆ R is (R, ≤∗)-generic over L(R). The upshot is that if we show that P(ω × ω)/Fin⊗2 is forcing equivalent to some topological Ramsey space, then (with minor modifcations to their proofs) the above theorem implies that the generic ultrafilter G2 has ‘complete combinatorics’.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 11 / 43

The structure behind E2: (ω ↓≤2, ≺)

Let ω ↓≤2 denote the set of non-decreasing sequences of members of ω of length less than or equal to 2.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 12 / 43

The structure behind E2: (ω ↓≤2, ≺)

Let ω ↓≤2 denote the set of non-decreasing sequences of members of ω of length less than or equal to 2.

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

Figure: ω ↓≤2

The well-order (ω ↓≤2, ≺) begins as follows: () ≺ (0) ≺ (0, 0) ≺ (0, 1) ≺ (1) ≺ (1, 1) ≺ (0, 2) ≺ (1, 2) ≺ (2) ≺ (2, 2) ≺

Dobrinen Higher dimensional Ellentuck spaces University of Denver 12 / 43

Constructing the maximal member of E2

∅ 18 19 12 17 13 {7} 16 11 8 3 15 10 6 4 14 9 5 2 1

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

∅ 1 2 3 4 5 6 7 8 () ≺ (0) ≺ (0, 0) ≺ (0, 1) ≺ (1) ≺ (1, 1) ≺ (0, 2) ≺ (1, 2) ≺ (2) ≺ (2, 2) ≺

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∅ {18} { 1 8 , 1 9 } {12} { 1 2 , 1 7 } { 1 2 , 1 3 } {7} { 7 , 1 6 } { 7 , 1 1 } { 7 , 8 } {3} { 3 , 1 5 } { 3 , 1 } { 3 , 6 } { 3 , 4 } {0} { , 1 4 } { , 9 } { , 5 } { , 2 } { , 1 }

Figure: Maximum element W2 ⊆ [ω]2 of E2

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

Figure: ω ↓≤2

∅ 1 2 3 4 5 6 7 8 () ≺ (0) ≺ (0, 0) ≺ (0, 1) ≺ (1) ≺ (1, 1) ≺ (0, 2) ≺ (1, 2) ≺ (2) ≺ (2, 2) ≺

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The space E2

∅ {18} { 1 8 , 1 9 } {12} { 1 2 , 1 7 } { 1 2 , 1 3 } {7} { 7 , 1 6 } { 7 , 1 1 } { 7 , 8 } {3} { 3 , 1 5 } { 3 , 1 } { 3 , 6 } { 3 , 4 } {0} { , 1 4 } { , 9 } { , 5 } { , 2 } { , 1 }

Figure: W2 ⊆ [ω]2

Dobrinen Higher dimensional Ellentuck spaces University of Denver 15 / 43

The space E2

∅ {18} { 1 8 , 1 9 } {12} { 1 2 , 1 7 } { 1 2 , 1 3 } {7} { 7 , 1 6 } { 7 , 1 1 } { 7 , 8 } {3} { 3 , 1 5 } { 3 , 1 } { 3 , 6 } { 3 , 4 } {0} { , 1 4 } { , 9 } { , 5 } { , 2 } { , 1 }

Figure: W2 ⊆ [ω]2

X ∈ E2 iff X is a subset of W2 such that (1) ˆ X is tree-isomorphic to W2, and (2) max values of the nodes of ˆ X are strictly increasing according to the wellordering ≺.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 15 / 43

The space E2

∅ {18} { 1 8 , 1 9 } {12} { 1 2 , 1 7 } { 1 2 , 1 3 } {7} { 7 , 1 6 } { 7 , 1 1 } { 7 , 8 } {3} { 3 , 1 5 } { 3 , 1 } { 3 , 6 } { 3 , 4 } {0} { , 1 4 } { , 9 } { , 5 } { , 2 } { , 1 }

Figure: W2 ⊆ [ω]2

X ∈ E2 iff X is a subset of W2 such that (1) ˆ X is tree-isomorphic to W2, and (2) max values of the nodes of ˆ X are strictly increasing according to the wellordering ≺. Note that lexicographic o.t.(X) = ω2 for each X ∈ E2.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 15 / 43

The space E2

∅ {18} { 1 8 , 1 9 } {12} { 1 2 , 1 7 } { 1 2 , 1 3 } {7} { 7 , 1 6 } { 7 , 1 1 } { 7 , 8 } {3} { 3 , 1 5 } { 3 , 1 } { 3 , 6 } { 3 , 4 } {0} { , 1 4 } { , 9 } { , 5 } { , 2 } { , 1 }

Figure: W2 ⊆ [ω]2

X ∈ E2 iff X is a subset of W2 such that (1) ˆ X is tree-isomorphic to W2, and (2) max values of the nodes of ˆ X are strictly increasing according to the wellordering ≺. Note that lexicographic o.t.(X) = ω2 for each X ∈ E2. Y ≤ X iff Y ⊆ X.

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Typical finite approximations to members of E2

∅ {18} { 1 8 , 1 9 } {7} { 7 , 1 6 } { 7 , 1 1 } {0} { , 3 5 } { , 1 4 } { , 5 } { , 1 }

Figure: r7(X)

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Typical finite approximations to members of E2

∅ {18} { 1 8 , 1 9 } {7} { 7 , 1 6 } { 7 , 1 1 } {0} { , 3 5 } { , 1 4 } { , 5 } { , 1 }

Figure: r7(X)

∅ {52} { 5 2 , 6 2 } {33} { 3 3 , 4 1 } { 3 3 , 3 4 } {18} { 1 8 , 3 9 } { 1 8 , 3 1 } { 1 8 , 1 9 } {3} { 3 , 3 6 } { 3 , 2 8 } { 3 , 1 } { 3 , 6 }

Figure: r10(Y )

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Why the funny ordering ≺?

Dobrinen Higher dimensional Ellentuck spaces University of Denver 17 / 43

Why the funny ordering ≺? It is necessary.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 17 / 43

Why the funny ordering ≺? It is necessary. In order to satisfy the Amalgamation Axiom (A3 (2)) in Todorcevic’s characteriztion of topological Ramsey spaces, some such requirement is necessary.

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(E2, ⊆, r) is a topological Ramsey space

• Thm. [D] (E2, ⊆, r) is a topological Ramsey space.

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(E2, ⊆, r) is a topological Ramsey space

• Thm. [D] (E2, ⊆, r) is a topological Ramsey space.

Thus, every subset of E2 with the property of Baire is Ramsey.

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(E2, ⊆, r) is a topological Ramsey space

• Thm. [D] (E2, ⊆, r) is a topological Ramsey space.

Thus, every subset of E2 with the property of Baire is Ramsey.

• Def. A set X ⊆ E2 is Ramsey iff for each basic open [a, X], there is a

Y ∈ [a, X] such that either [a, Y ] ⊆ X or [a, Y ] ∩ X = ∅. AR denotes the collection of all finite approximations of members of E2. For a ∈ AR and X ∈ E2, [a, X] := {Y ∈ E2 : a ❁ Y ⊆ X}. The Ellentuck topology is generated by basic open sets of the form [a, X], where a ∈ AR and X ∈ E2.

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P(ω × ω)/Fin⊗2 is forcing equivalent to a new topological Ramsey space

(E2, ⊆Fin

⊗2

) is forcing equivalent to ((Fin⊗2)+, ⊆Fin

⊗2

).

Dobrinen Higher dimensional Ellentuck spaces University of Denver 19 / 43

P(ω × ω)/Fin⊗2 is forcing equivalent to a new topological Ramsey space

(E2, ⊆Fin

⊗2

) is forcing equivalent to ((Fin⊗2)+, ⊆Fin

⊗2

). (Below any member A ∈ (Fin⊗2)+ is some B ⊆ A which is an isomorphic copy of W2, and below B, there is a dense subset of (Fin⊗2)+ ↾ B isomorphic to E2.)

Dobrinen Higher dimensional Ellentuck spaces University of Denver 19 / 43

Higher order forcings

Fin⊗3 is the ideal on ω × ω × ω such that X ⊆ ω3 is in Fin⊗3 iff for all but finitely many i < ω, the i-th fiber of X, {(j, k) ∈ ω × ω : (i, j, k) ∈ X}, is in Fin ⊗ Fin.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 20 / 43

Higher order forcings

Fin⊗3 is the ideal on ω × ω × ω such that X ⊆ ω3 is in Fin⊗3 iff for all but finitely many i < ω, the i-th fiber of X, {(j, k) ∈ ω × ω : (i, j, k) ∈ X}, is in Fin ⊗ Fin. P(ω3)/Fin⊗3 adds a generic ultrafilter G3 on ω3 such that its projection to the first two coordinates is a generic ultrafilter forced by P(ω2)/Fin⊗2, and its projection to the first coordinate is a Ramsey ultrafilter forced by P(ω)/Fin.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 20 / 43

Higher order forcings

Fin⊗3 is the ideal on ω × ω × ω such that X ⊆ ω3 is in Fin⊗3 iff for all but finitely many i < ω, the i-th fiber of X, {(j, k) ∈ ω × ω : (i, j, k) ∈ X}, is in Fin ⊗ Fin. P(ω3)/Fin⊗3 adds a generic ultrafilter G3 on ω3 such that its projection to the first two coordinates is a generic ultrafilter forced by P(ω2)/Fin⊗2, and its projection to the first coordinate is a Ramsey ultrafilter forced by P(ω)/Fin. We thin (Fin⊗3)+ to a topological Ramsey space E3 forcing equivalent (when partially ordered by ⊆Fin

⊗3

) to P(ω3)/Fin⊗3 .

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The structure behind E3

The well-order (ω ↓≤3, ≺) begins as follows: ∅ ≺ (0) ≺ (0, 0) ≺ (0, 0, 0) ≺ (0, 0, 1) ≺ (0, 1) ≺ (0, 1, 1) ≺ (1) ≺ (1, 1) ≺ (1, 1, 1) ≺ (0, 0, 2) ≺ (0, 1, 2) ≺ (0, 2) ≺ (0, 2, 2) ≺ (1, 1, 2) ≺ (1, 2) ≺ (1, 2, 2) ≺ (2) ≺ (2, 2) ≺ (2, 2, 2) ≺ (0, 0, 3) ≺ · · · (1)

∅ (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: ω ↓≤3

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∅ {31} {31, 32} {31,32,33} {16} {16, 29} {16,29,30} {16, 17} {16,17,28} {16,17,18} {6} {6, 26} {6,26,27} {6, 14} {6,14,25} {6,14,15} {6, 7} {6,7,24} {6,7,13} {6,7,8} {0} {0, 22} {0,22,23} {0, 11} {0,11,21} {0,11,12} {0, 4} {0,4,20} {0,4,10} {0,4,5} {0, 1} {0,1,19} {0,1,9} {0,1,3} {0,1,2}

Figure: The maximum member of E3, W3 ⊆ [ω]3

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∅ {31} {31, 32} {31,32,33} {16} {16, 29} {16,29,30} {16, 17} {16,17,28} {16,17,18} {6} {6, 26} {6,26,27} {6, 14} {6,14,25} {6,14,15} {6, 7} {6,7,24} {6,7,13} {6,7,8} {0} {0, 22} {0,22,23} {0, 11} {0,11,21} {0,11,12} {0, 4} {0,4,20} {0,4,10} {0,4,5} {0, 1} {0,1,19} {0,1,9} {0,1,3} {0,1,2}

Figure: The maximum member of E3, W3 ⊆ [ω]3

∅ (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: ω ↓≤3

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The space E3

∅ {31} {31, 32} {31,32,33} {16} {16, 29} {16,29,30} {16, 17} {16,17,28} {16,17,18} {6} {6, 26} {6,26,27} {6, 14} {6,14,25} {6,14,15} {6, 7} {6,7,24} {6,7,13} {6,7,8} {0} {0, 22} {0,22,23} {0, 11} {0,11,21} {0,11,12} {0, 4} {0,4,20} {0,4,10} {0,4,5} {0, 1} {0,1,19} {0,1,9} {0,1,3} {0,1,2}

Figure: W3

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The space E3

∅ {31} {31, 32} {31,32,33} {16} {16, 29} {16,29,30} {16, 17} {16,17,28} {16,17,18} {6} {6, 26} {6,26,27} {6, 14} {6,14,25} {6,14,15} {6, 7} {6,7,24} {6,7,13} {6,7,8} {0} {0, 22} {0,22,23} {0, 11} {0,11,21} {0,11,12} {0, 4} {0,4,20} {0,4,10} {0,4,5} {0, 1} {0,1,19} {0,1,9} {0,1,3} {0,1,2}

Figure: W3

X ∈ E3 iff X ⊆ W3 and X ∼ = W3 as a tree, and also with respect to the ≺

• rder of the node labels.

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The space E3

∅ {31} {31, 32} {31,32,33} {16} {16, 29} {16,29,30} {16, 17} {16,17,28} {16,17,18} {6} {6, 26} {6,26,27} {6, 14} {6,14,25} {6,14,15} {6, 7} {6,7,24} {6,7,13} {6,7,8} {0} {0, 22} {0,22,23} {0, 11} {0,11,21} {0,11,12} {0, 4} {0,4,20} {0,4,10} {0,4,5} {0, 1} {0,1,19} {0,1,9} {0,1,3} {0,1,2}

Figure: W3

X ∈ E3 iff X ⊆ W3 and X ∼ = W3 as a tree, and also with respect to the ≺

• rder of the node labels.

Y ≤ X iff Y ⊆ X.

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∅ {31} {31, 32} { 3 1 , 3 2 , 3 3 } {0} {0, 36} { , 3 6 , 3 7 } {0, 11} { , 1 1 , 3 5 } { , 1 1 , 2 1 } {0, 1} { , 1 , 3 4 } { , 1 , 9 } { , 1 , 2 }

Figure: r7(Y ), a typical finite approximation to a member of E3

() (1) (1, 1) ( 1 , 1 , 1 ) (0) (0, 2) ( , 2 , 2 ) (0, 1) ( , 1 , 2 ) ( , 1 , 1 ) (0, 0) ( , , 2 ) ( , , 1 ) ( , , ) Dobrinen Higher dimensional Ellentuck spaces University of Denver 24 / 43

We now define the spaces Ek, k ≥ 2, in general.

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The well-ordered set (ω ↓≤k, ≺), k ≥ 2.

ω ↓≤k denotes the collection of all non-decreasing sequences of members of ω of length less than or equal to k.

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The well-ordered set (ω ↓≤k, ≺), k ≥ 2.

ω ↓≤k denotes the collection of all non-decreasing sequences of members of ω of length less than or equal to k. Define a well-ordering ≺ on ω ↓≤k as follows: () is the ≺-minimum element. For (j0, . . . , jp−1) and (l0, . . . , lq−1) in ω ↓≤k with p, q ≥ 1, define (j0, . . . , jp−1) ≺ (l0, . . . , lq−1) if and only if either

1 jp−1 < lq−1, or 2 jp−1 = lq−1 and (j0, . . . , jp−1) <lex (l0, . . . , lq−1). Dobrinen Higher dimensional Ellentuck spaces University of Denver 26 / 43

The well-ordered set (ω ↓≤k, ≺), k ≥ 2.

ω ↓≤k denotes the collection of all non-decreasing sequences of members of ω of length less than or equal to k. Define a well-ordering ≺ on ω ↓≤k as follows: () is the ≺-minimum element. For (j0, . . . , jp−1) and (l0, . . . , lq−1) in ω ↓≤k with p, q ≥ 1, define (j0, . . . , jp−1) ≺ (l0, . . . , lq−1) if and only if either

1 jp−1 < lq−1, or 2 jp−1 = lq−1 and (j0, . . . , jp−1) <lex (l0, . . . , lq−1).

Let jm denote the ≺ −m-th member of ω ↓≤k. For l ∈ ω ↓≤k, we let m

l ∈ ω denote the m such that

l = jm.

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The spaces Ek, k ≥ 2

• Wk is the image of the function

l → {m : jm ⊑ l}, l ∈ ω ↓≤k.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 27 / 43

The spaces Ek, k ≥ 2

• Wk is the image of the function

l → {m : jm ⊑ l}, l ∈ ω ↓≤k. We say that X is an Ek-tree if X is a function from ω ↓≤k into Wk such that (i) For each m < ω, X( jm) ∈ [ω]|

jm| ∩

Wk; (ii) For all 1 ≤ m < ω, max( X( jm)) < max( X( jm+1)); (iii) For all m, n < ω, X( jm) ❁ X( jn) if and only if jm ❁ jn.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 27 / 43

The spaces Ek, k ≥ 2

• Wk is the image of the function

l → {m : jm ⊑ l}, l ∈ ω ↓≤k. We say that X is an Ek-tree if X is a function from ω ↓≤k into Wk such that (i) For each m < ω, X( jm) ∈ [ω]|

jm| ∩

Wk; (ii) For all 1 ≤ m < ω, max( X( jm)) < max( X( jm+1)); (iii) For all m, n < ω, X( jm) ❁ X( jn) if and only if jm ❁ jn. The space Ek consists of all X := [ X], where X is an Ek-tree.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 27 / 43

The spaces Ek, k ≥ 2

• Wk is the image of the function

l → {m : jm ⊑ l}, l ∈ ω ↓≤k. We say that X is an Ek-tree if X is a function from ω ↓≤k into Wk such that (i) For each m < ω, X( jm) ∈ [ω]|

jm| ∩

Wk; (ii) For all 1 ≤ m < ω, max( X( jm)) < max( X( jm+1)); (iii) For all m, n < ω, X( jm) ❁ X( jn) if and only if jm ❁ jn. The space Ek consists of all X := [ X], where X is an Ek-tree. For X, Y ∈ Ek, Y ≤ X iff Y ⊆ X.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 27 / 43

The spaces Ek, k ≥ 2

• Wk is the image of the function

l → {m : jm ⊑ l}, l ∈ ω ↓≤k. We say that X is an Ek-tree if X is a function from ω ↓≤k into Wk such that (i) For each m < ω, X( jm) ∈ [ω]|

jm| ∩

Wk; (ii) For all 1 ≤ m < ω, max( X( jm)) < max( X( jm+1)); (iii) For all m, n < ω, X( jm) ❁ X( jn) if and only if jm ❁ jn. The space Ek consists of all X := [ X], where X is an Ek-tree. For X, Y ∈ Ek, Y ≤ X iff Y ⊆ X. For each n < ω, the n-th finite aproximation rn(X) is X ∩ ({ ip : p < n} × Wk), where ( ip : p < ω) is the ≺-wellordering on ω ↓k.

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The Ek are high dimensional Ellentuck spaces

• Thm. [D] For each 2 ≤ k < ω, (Ek, ⊆, r) is a topological Ramsey space.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 28 / 43

The Ek are high dimensional Ellentuck spaces

• Thm. [D] For each 2 ≤ k < ω, (Ek, ⊆, r) is a topological Ramsey space.

Remarks.

1 Each space Ek+1 is comprised of ω many copies of Ek. Dobrinen Higher dimensional Ellentuck spaces University of Denver 28 / 43

The Ek are high dimensional Ellentuck spaces

• Thm. [D] For each 2 ≤ k < ω, (Ek, ⊆, r) is a topological Ramsey space.

Remarks.

1 Each space Ek+1 is comprised of ω many copies of Ek. 2 Moreover, each projection of Ek to levels 1 through j produces a

copy of Ej.

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The Ek are high dimensional Ellentuck spaces

• Thm. [D] For each 2 ≤ k < ω, (Ek, ⊆, r) is a topological Ramsey space.

Remarks.

1 Each space Ek+1 is comprised of ω many copies of Ek. 2 Moreover, each projection of Ek to levels 1 through j produces a

copy of Ej.

3 The trick was finding the right thinning and finite approximation

scheme to make Axiom A.3 (2) hold. (The Pigeonhole Principle A.4 was no problem.)

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Initial Tukey and Rudin-Keisler structures below Gk, k ≥ 2

• Thm. [D] Let Gk denote the generic ultrafilter forced by P(ωk)/Fin⊗k.

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Initial Tukey and Rudin-Keisler structures below Gk, k ≥ 2

• Thm. [D] Let Gk denote the generic ultrafilter forced by P(ωk)/Fin⊗k.

1 If V ≤T Gk, then V ≡T πl(Gk) for some l ≤ k. Dobrinen Higher dimensional Ellentuck spaces University of Denver 29 / 43

Initial Tukey and Rudin-Keisler structures below Gk, k ≥ 2

• Thm. [D] Let Gk denote the generic ultrafilter forced by P(ωk)/Fin⊗k.

1 If V ≤T Gk, then V ≡T πl(Gk) for some l ≤ k. 2 Thus, the Tukey equivalence classes of (nonprincipal) ultrafilters

Tukey reducible to Gk form a chain of length k.

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Initial Tukey and Rudin-Keisler structures below Gk, k ≥ 2

• Thm. [D] Let Gk denote the generic ultrafilter forced by P(ωk)/Fin⊗k.

1 If V ≤T Gk, then V ≡T πl(Gk) for some l ≤ k. 2 Thus, the Tukey equivalence classes of (nonprincipal) ultrafilters

Tukey reducible to Gk form a chain of length k.

3 Further, the Rudin-Keisler equivalence classes of (nonprincipal)

ultrafilters RK-reducible to Gk form a chain of length k.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 29 / 43

Initial Tukey and Rudin-Keisler structures below Gk, k ≥ 2

• Thm. [D] Let Gk denote the generic ultrafilter forced by P(ωk)/Fin⊗k.

1 If V ≤T Gk, then V ≡T πl(Gk) for some l ≤ k. 2 Thus, the Tukey equivalence classes of (nonprincipal) ultrafilters

Tukey reducible to Gk form a chain of length k.

3 Further, the Rudin-Keisler equivalence classes of (nonprincipal)

ultrafilters RK-reducible to Gk form a chain of length k.

• Remark. The fact that Ek is dense below any member of (Fin⊗k)+

provides a simple way of reading off the partition relations for the generic ultrafilter.

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Outline of Proof

1 Show (Ek, ⊆Fin ⊗k

) is forcing equivalent to P(ωk)/Fin⊗k.

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Outline of Proof

1 Show (Ek, ⊆Fin ⊗k

) is forcing equivalent to P(ωk)/Fin⊗k.

2 Prove (Ek, ≤, r) is a topological Ramsey space. Dobrinen Higher dimensional Ellentuck spaces University of Denver 30 / 43

Outline of Proof

1 Show (Ek, ⊆Fin ⊗k

) is forcing equivalent to P(ωk)/Fin⊗k.

2 Prove (Ek, ≤, r) is a topological Ramsey space. 3 Prove a Ramsey-classification theorem for equivalence relations on

fronts on Ek, extending the Pudl´ ak-R¨

• dl Theorem for the Ellentuck

space.

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Outline of Proof

1 Show (Ek, ⊆Fin ⊗k

) is forcing equivalent to P(ωk)/Fin⊗k.

2 Prove (Ek, ≤, r) is a topological Ramsey space. 3 Prove a Ramsey-classification theorem for equivalence relations on

fronts on Ek, extending the Pudl´ ak-R¨

• dl Theorem for the Ellentuck

space.

4 Prove Basic Cofinal Maps Theorem, the correct analogue for our

spaces of ‘every p-point having continuous Tukey reductions’.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 30 / 43

Outline of Proof

1 Show (Ek, ⊆Fin ⊗k

) is forcing equivalent to P(ωk)/Fin⊗k.

2 Prove (Ek, ≤, r) is a topological Ramsey space. 3 Prove a Ramsey-classification theorem for equivalence relations on

fronts on Ek, extending the Pudl´ ak-R¨

• dl Theorem for the Ellentuck

space.

4 Prove Basic Cofinal Maps Theorem, the correct analogue for our

spaces of ‘every p-point having continuous Tukey reductions’.

5 For V ≤T Gk, apply Basic Cofinal Maps Theorem to find a front F on

Ek and an f : F → ω such that V = f (Gk|F).

Dobrinen Higher dimensional Ellentuck spaces University of Denver 30 / 43

Outline of Proof

1 Show (Ek, ⊆Fin ⊗k

) is forcing equivalent to P(ωk)/Fin⊗k.

2 Prove (Ek, ≤, r) is a topological Ramsey space. 3 Prove a Ramsey-classification theorem for equivalence relations on

fronts on Ek, extending the Pudl´ ak-R¨

• dl Theorem for the Ellentuck

space.

4 Prove Basic Cofinal Maps Theorem, the correct analogue for our

spaces of ‘every p-point having continuous Tukey reductions’.

5 For V ≤T Gk, apply Basic Cofinal Maps Theorem to find a front F on

Ek and an f : F → ω such that V = f (Gk|F).

6 Apply the Ramsey-classification theorem for equivalence relations on

fronts and analyze f (Gk|F).

Dobrinen Higher dimensional Ellentuck spaces University of Denver 30 / 43

• Def. A family of finite approximations F is a front on Ek iff

(i) ∀X ∈ Ek, ∃a ∈ F such that a ❁ X; and (ii) for a, b ∈ F, a ❁ b.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 31 / 43

• Def. A family of finite approximations F is a front on Ek iff

(i) ∀X ∈ Ek, ∃a ∈ F such that a ❁ X; and (ii) for a, b ∈ F, a ❁ b.

• Def. A map ϕ on a front F ⊆ AR is called

1 inner if for each a ∈ F, ϕ(a) is a subtree of

a.

2 Nash-Williams if for all pairs a, b ∈ F, ϕ(a) = ϕ(b) implies

ϕ(a) ❁ ϕ(b) (in terms of r).

3 irreducible if it is inner and Nash-Williams. Dobrinen Higher dimensional Ellentuck spaces University of Denver 31 / 43

Ramsey-classification Theorem for equivalence relations on fronts

• Thm. [D] Let F be a front on Ek and f : F → ω. Then there exists an

X ∈ Ek and an irreducible map ϕ on F|X such that for all a, b ∈ F|X, f (a) = f (b) iff ϕ(a) = ϕ(b).

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Ramsey-classification Theorem for equivalence relations on fronts

• Thm. [D] Let F be a front on Ek and f : F → ω. Then there exists an

X ∈ Ek and an irreducible map ϕ on F|X such that for all a, b ∈ F|X, f (a) = f (b) iff ϕ(a) = ϕ(b).

• Rem. This is the analogue (extension) of the Pudl´

ak-R¨

• dl Theorem for

this space. Further, the canonization maps have the form that ϕ(a) is a projection to some initial segements of the nodes in a.

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Ramsey-classification Theorem for equivalence relations on fronts

• Thm. [D] Let F be a front on Ek and f : F → ω. Then there exists an

X ∈ Ek and an irreducible map ϕ on F|X such that for all a, b ∈ F|X, f (a) = f (b) iff ϕ(a) = ϕ(b).

• Rem. This is the analogue (extension) of the Pudl´

ak-R¨

• dl Theorem for

this space. Further, the canonization maps have the form that ϕ(a) is a projection to some initial segements of the nodes in a.

• Thm. [D] Let R be an equivalence relation on some front F on Ek.

Suppose ϕ and ϕ′ are irreducible maps canonizing R. Then there is an A ∈ Ek such that for each a ∈ F|A, ϕ(a) = ϕ′(a).

Dobrinen Higher dimensional Ellentuck spaces University of Denver 32 / 43

For a front F consisting of the n-th finite approximations ARn, the canonical equivalence relations are given by projection maps of the form ϕ(a(0), . . . , a(n − 1)) = (πj0(a(0)), . . . , πjn−1(a(n − 1))), where πj(a(i)) is the projection of a(i) to its first j levels (in the tree Wk).

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Basic Cofinal Maps from Gk

• Def. Given Y ∈ Bk := Gk ∩ Ek, a monotone map g : Bk|Y → P(ω) is

basic if there is a map ˆ g : AR|Y → [ω]<ω such that

1 (monotonicity) For all s, t ∈ AR|Y , s ⊆ t → ˆ

g(s) ⊆ ˆ g(t);

2 (initial segment preserving) For s ❁ t in AR|Y , ˆ

g(s) ⊑ ˆ g(t);

3 (ˆ

g represents g) For each V ∈ Bk|Y , g(V ) =

n<ω ˆ

g(rn(V )).

Dobrinen Higher dimensional Ellentuck spaces University of Denver 34 / 43

Basic Cofinal Maps from Gk

• Def. Given Y ∈ Bk := Gk ∩ Ek, a monotone map g : Bk|Y → P(ω) is

basic if there is a map ˆ g : AR|Y → [ω]<ω such that

1 (monotonicity) For all s, t ∈ AR|Y , s ⊆ t → ˆ

g(s) ⊆ ˆ g(t);

2 (initial segment preserving) For s ❁ t in AR|Y , ˆ

g(s) ⊑ ˆ g(t);

3 (ˆ

g represents g) For each V ∈ Bk|Y , g(V ) =

n<ω ˆ

g(rn(V )).

• Thm. (Basic monotone maps on Gk) [D]

Let Gk generic for P(ωk)/Fin⊗k. In V [Gk], for each monotone function g : Gk → P(ω), there is a Y ∈ Bk such that g ↾ (Bk|Y ) is basic.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 34 / 43

• Remark. The proofs of the Ramsey-classification Theorem for

equivalence relations on fronts and the Basic Cofinal Maps Theorem could be proved using only the Abstract Nash-Williams Theorem, which we originally proved without using A.3 (2).

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Infinite dimensional Ellentuck spaces

The sets [ω]k are actually uniform barriers (on ω) of finite rank. Uniform barriers B (on ω) of any countably infinite rank provide the template for building higher order Ellentuck spaces EB. Such spaces EB are forcing equivalent to forcings constructed by continuing the process of iteratively constructing ideals built from the ideals Fin⊗k. Rather than give all the definitions, we shall now provide an example giving the flavor of these spaces.

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The first infinite dimensional Ellentuck space

Let S denote {a ∈ [ω]<ω : |a| = min(a) + 1}. S is the Schreier barrier.

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The first infinite dimensional Ellentuck space

Let S denote {a ∈ [ω]<ω : |a| = min(a) + 1}. S is the Schreier barrier. FinS is an ideal on S: X ⊆ S is in FinS iff for all but finitely many k < ω, {a \ {k} : a ∈ X and min(a) = k} ∈ Fin⊗k.

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The first infinite dimensional Ellentuck space

Let S denote {a ∈ [ω]<ω : |a| = min(a) + 1}. S is the Schreier barrier. FinS is an ideal on S: X ⊆ S is in FinS iff for all but finitely many k < ω, {a \ {k} : a ∈ X and min(a) = k} ∈ Fin⊗k. X ⊆ S is in (FinS)+ iff there are infinitely many k such that {a \ {k} : a ∈ X and min(a) = k} ∈ (Fin⊗k)+.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 37 / 43

The first infinite dimensional Ellentuck space

Let S denote {a ∈ [ω]<ω : |a| = min(a) + 1}. S is the Schreier barrier. FinS is an ideal on S: X ⊆ S is in FinS iff for all but finitely many k < ω, {a \ {k} : a ∈ X and min(a) = k} ∈ Fin⊗k. X ⊆ S is in (FinS)+ iff there are infinitely many k such that {a \ {k} : a ∈ X and min(a) = k} ∈ (Fin⊗k)+. P(S)/FinS is forcing equivalent to ((FinS)+, ⊆Fin

S

).

Dobrinen Higher dimensional Ellentuck spaces University of Denver 37 / 43

We use the form of S to make our template structure of finite non-decreasing sequences of natural numbers.

∅ (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: ω ↓S

() ≺ (0) ≺ (1) ≺ (1, 1) ≺ (1, 2) ≺ (2) ≺ (2, 2) ≺ (2, 2, 2) ≺ (1, 3) ≺ (2, 2, 3) ≺ (2, 3) ≺ (2, 3, 3) ≺ (3) ≺ (3, 3) ≺ (3, 3, 3) ≺ (3, 3, 3, 3) ≺ (1, 4) ≺ . . .

Dobrinen Higher dimensional Ellentuck spaces University of Denver 38 / 43

∅ 11 23 24 25 12 21 22 13 20 14 4 18 19 9 17 10 5 16 8 6 1 15 7 3 2

Figure: WS

∅ (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: ω ↓S

Dobrinen Higher dimensional Ellentuck spaces University of Denver 39 / 43

∅ {11} {11, 23} {11, 23, 24} {11,23,24,25} {11, 12} {11, 12, 21} {11,12,21,22} {11, 12, 13} {11,12,13,20} {11,12,13,14} {4} {4, 18} {4,18,19} {4, 9} {4,9,17} {4,9,10} {4, 5} {4,5,16} {4,5,8} {4,5,6} {1} {1, 15} {1, 7} {1, 3} {1, 2} {0}

Figure: WS

∅ (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: ω ↓S

Dobrinen Higher dimensional Ellentuck spaces University of Denver 40 / 43

The space ES, for S the Schreier barrier

ES is the collection of all X ⊆ WS such that

1 for infinitely many k, {a \ {k} : a ∈ X and min a = k} ∈ Ek, 2 if {a ∈ X : min a = k} ∈ Ek, then it is empty, 3 The values of the nodes in ˆ

X follow the ≺ order.

4 Finitization is recursively induced by the finitizations on the Ek.

∅ {11} {11, 23} {11, 23, 24} {11,23,24,25} {11, 12} {11, 12, 21} {11,12,21,22} {11, 12, 13} {11,12,13,20} {11,12,13,14} {4} {4, 18} {4,18,19} {4, 9} {4,9,17} {4,9,10} {4, 5} {4,5,16} {4,5,8} {4,5,6} {1} {1, 15} {1, 7} {1, 3} {1, 2} {0}

Figure: WS

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Current work

Let B be any uniform barrier on ω.

• Thm. [D] The space EB is a topological Ramsey space.

Forcing with (EB, ⊆Fin

B

) is equivalent to forcing with P(B)/FinB.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 42 / 43

Current work

Let B be any uniform barrier on ω.

• Thm. [D] The space EB is a topological Ramsey space.

Forcing with (EB, ⊆Fin

B

) is equivalent to forcing with P(B)/FinB.

• Thm. [D]

1 Equivalence relations on AR1 are canonized as uniform fronts on

WB; that is, projections which have the form of a uniform front.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 42 / 43

Current work

Let B be any uniform barrier on ω.

• Thm. [D] The space EB is a topological Ramsey space.

Forcing with (EB, ⊆Fin

B

) is equivalent to forcing with P(B)/FinB.

• Thm. [D]

1 Equivalence relations on AR1 are canonized as uniform fronts on

WB; that is, projections which have the form of a uniform front.

2 The initial Rudin-Keisler structure below the generic ultrafilter GB

is the linear ordering of the GB-equivalence classes of the uniform fronts on WB.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 42 / 43

Current work

Let B be any uniform barrier on ω.

• Thm. [D] The space EB is a topological Ramsey space.

Forcing with (EB, ⊆Fin

B

) is equivalent to forcing with P(B)/FinB.

• Thm. [D]

1 Equivalence relations on AR1 are canonized as uniform fronts on

WB; that is, projections which have the form of a uniform front.

2 The initial Rudin-Keisler structure below the generic ultrafilter GB

is the linear ordering of the GB-equivalence classes of the uniform fronts on WB.

3 Special Case: For the Schreier barrier S, the initial Rudin-Keisler

structure below the generic ultrafilter GS is the ultrapower of N modulo the projected Ramsey ultrafilter π1(GS).

Dobrinen Higher dimensional Ellentuck spaces University of Denver 42 / 43

Current work

Let B be any uniform barrier on ω.

• Thm. [D] The space EB is a topological Ramsey space.

Forcing with (EB, ⊆Fin

B

) is equivalent to forcing with P(B)/FinB.

• Thm. [D]

1 Equivalence relations on AR1 are canonized as uniform fronts on

WB; that is, projections which have the form of a uniform front.

2 The initial Rudin-Keisler structure below the generic ultrafilter GB

is the linear ordering of the GB-equivalence classes of the uniform fronts on WB.

3 Special Case: For the Schreier barrier S, the initial Rudin-Keisler

structure below the generic ultrafilter GS is the ultrapower of N modulo the projected Ramsey ultrafilter π1(GS).

Dobrinen Higher dimensional Ellentuck spaces University of Denver 42 / 43

• Thm. [D]

1 We have Ramsey-classification theorems canonizing equivalence

relations on barriers on EB in terms of irreducible functions.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 43 / 43

• Thm. [D]

1 We have Ramsey-classification theorems canonizing equivalence

relations on barriers on EB in terms of irreducible functions.

2 The initial Tukey structure below GB has cardinality c, and

contains the linear order of the GB equivalence classes of the uniform fronts on WB.

Dobrinen Higher dimensional Ellentuck spaces University of Denver 43 / 43

• Thm. [D]

1 We have Ramsey-classification theorems canonizing equivalence

relations on barriers on EB in terms of irreducible functions.

2 The initial Tukey structure below GB has cardinality c, and

contains the linear order of the GB equivalence classes of the uniform fronts on WB. Work in progress: Double checking the proofs, finding the exact initial Tukey structures and RK classes within (Is (2) above exact?).

Dobrinen Higher dimensional Ellentuck spaces University of Denver 43 / 43