⊗ 2 P ( ω × ω ) / Fin ⊗ 2 is forcing equivalent to ((Fin ⊗ Fin) + , ⊆ Fin ), Dobrinen Higher dimensional Ellentuck spaces University of Denver 7 / 43
⊗ 2 P ( ω × ω ) / Fin ⊗ 2 is forcing equivalent to ((Fin ⊗ Fin) + , ⊆ Fin ), which is forcing equivalent to { X ⊆ ω × ω : infinitely many fibers of X are infinite, and all finite fibers of ⊗ 2 X are empty } , partially ordered by ⊆ Fin . Dobrinen Higher dimensional Ellentuck spaces University of Denver 7 / 43
⊗ 2 P ( ω × ω ) / Fin ⊗ 2 is forcing equivalent to ((Fin ⊗ Fin) + , ⊆ Fin ), which is forcing equivalent to { X ⊆ ω × ω : infinitely many fibers of X are infinite, and all finite fibers of ⊗ 2 X are empty } , partially ordered by ⊆ Fin . 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 E 2 which is forcing equivalent to P ( ω × ω ) / Fin ⊗ 2 . Our space E 2 looks and acts like ω copies of the Ellentuck space, given a judiciously chosen finitization map. Dobrinen Higher dimensional Ellentuck spaces University of Denver 7 / 43
Review Dobrinen Higher dimensional Ellentuck spaces University of Denver 8 / 43
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 ( r n ( 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 ( r n ( 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 ( r n ( 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 ( r n ( 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 AR N ), 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 ( r n ( 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 AR N ), then ( R , ≤ , r ) is a topological Ramsey space. n -th Appproximations: AR n = { r n ( X ) : X ∈ R} . Finite Approximations: AR = � n <ω AR n . 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 ) . Dobrinen Higher dimensional Ellentuck spaces University of Denver 11 / 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 ) . 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 G 2 has ‘complete combinatorics’. Dobrinen Higher dimensional Ellentuck spaces University of Denver 11 / 43
The structure behind E 2 : ( 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 E 2 : ( ω � ↓≤ 2 , ≺ ) Let ω � ↓≤ 2 denote the set of non-decreasing sequences of members of ω of length less than or equal to 2. ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 0 1 2 3 4 1 2 3 4 2 3 4 3 4 4 , , , , , , , , , , , , , , , 0 0 0 0 0 1 1 1 1 2 2 2 3 3 4 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (0) (1) (2) (3) (4) () 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 E 2 1 2 5 9 14 4 6 10 15 8 11 16 13 17 19 0 3 { 7 } 12 18 ∅ ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 0 1 2 3 4 1 2 3 4 2 3 4 3 4 4 , , , , , , , , , , , , , , , 0 0 0 0 0 1 1 1 1 2 2 2 3 3 4 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (0) (1) (2) (3) (4) () ∅ 0 1 2 3 4 5 6 7 8 () ≺ (0) ≺ (0 , 0) ≺ (0 , 1) ≺ (1) ≺ (1 , 1) ≺ (0 , 2) ≺ (1 , 2) ≺ (2) ≺ (2 , 2) ≺ Dobrinen Higher dimensional Ellentuck spaces University of Denver 13 / 43
} } } } } } } } 3 7 9 } } } } 4 } } 0 5 } 1 6 1 1 1 1 2 5 9 1 4 6 1 1 8 1 1 , , , 2 2 8 , , , , , , , , , , , , 0 0 0 0 0 3 3 3 3 7 7 7 1 1 1 { { { { { { { { { { { { { { { { 0 } { 3 } { 7 } { 12 } { 18 } ∅ Figure: Maximum element W 2 ⊆ [ ω ] 2 of E 2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 0 1 2 3 4 1 2 3 4 2 3 4 3 4 4 , , , , , , , , , , , , , , , 0 0 0 0 0 1 1 1 1 2 2 2 3 3 4 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (0) (1) (2) (3) (4) () Figure: ω � ↓≤ 2 ∅ 0 1 2 3 4 5 6 7 8 () ≺ (0) ≺ (0 , 0) ≺ (0 , 1) ≺ (1) ≺ (1 , 1) ≺ (0 , 2) ≺ (1 , 2) ≺ (2) ≺ (2 , 2) ≺ Dobrinen Higher dimensional Ellentuck spaces University of Denver 14 / 43
The space E 2 } } } } } } } } 3 7 9 } } } } 4 } } 0 5 } 1 6 1 1 1 1 2 5 9 1 4 6 1 1 8 1 1 , , , 2 2 8 , , , , , , , , , , , , 0 0 0 0 0 3 3 3 3 7 7 7 1 1 1 { { { { { { { { { { { { { { { { 0 } { 3 } { 7 } { 12 } { 18 } ∅ Figure: W 2 ⊆ [ ω ] 2 Dobrinen Higher dimensional Ellentuck spaces University of Denver 15 / 43
The space E 2 } } } } } } } } 3 7 9 } } } } 4 } } 0 5 } 1 6 1 1 1 1 2 5 9 1 4 6 1 1 8 1 1 , , , 2 2 8 , , , , , , , , , , , , 0 0 0 0 0 3 3 3 3 7 7 7 1 1 1 { { { { { { { { { { { { { { { { 0 } { 3 } { 7 } { 12 } { 18 } ∅ Figure: W 2 ⊆ [ ω ] 2 X ∈ E 2 iff X is a subset of W 2 such that (1) ˆ X is tree-isomorphic to � W 2 , 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 E 2 } } } } } } } } 3 7 9 } } } } 4 } } 0 5 } 1 6 1 1 1 1 2 5 9 1 4 6 1 1 8 1 1 , , , 2 2 8 , , , , , , , , , , , , 0 0 0 0 0 3 3 3 3 7 7 7 1 1 1 { { { { { { { { { { { { { { { { 0 } { 3 } { 7 } { 12 } { 18 } ∅ Figure: W 2 ⊆ [ ω ] 2 X ∈ E 2 iff X is a subset of W 2 such that (1) ˆ X is tree-isomorphic to � W 2 , and (2) max values of the nodes of ˆ X are strictly increasing according to the wellordering ≺ . Note that lexicographic o.t.( X ) = ω 2 for each X ∈ E 2 . Dobrinen Higher dimensional Ellentuck spaces University of Denver 15 / 43
The space E 2 } } } } } } } } 3 7 9 } } } } 4 } } 0 5 } 1 6 1 1 1 1 2 5 9 1 4 6 1 1 8 1 1 , , , 2 2 8 , , , , , , , , , , , , 0 0 0 0 0 3 3 3 3 7 7 7 1 1 1 { { { { { { { { { { { { { { { { 0 } { 3 } { 7 } { 12 } { 18 } ∅ Figure: W 2 ⊆ [ ω ] 2 X ∈ E 2 iff X is a subset of W 2 such that (1) ˆ X is tree-isomorphic to � W 2 , and (2) max values of the nodes of ˆ X are strictly increasing according to the wellordering ≺ . Note that lexicographic o.t.( X ) = ω 2 for each X ∈ E 2 . Y ≤ X iff Y ⊆ X . Dobrinen Higher dimensional Ellentuck spaces University of Denver 15 / 43
Typical finite approximations to members of E 2 } } } } } 9 } } 4 5 1 6 1 1 5 1 3 1 1 , 8 , , , , , , 0 0 0 0 7 7 1 { { { { { { { { 0 } { 7 } { 18 } ∅ Figure: r 7 ( X ) Dobrinen Higher dimensional Ellentuck spaces University of Denver 16 / 43
Typical finite approximations to members of E 2 } } } } } 9 } } 4 5 1 6 1 1 5 1 3 1 1 , 8 , , , , , , 0 0 0 0 7 7 1 { { { { { { { { 0 } { 7 } { 18 } ∅ Figure: r 7 ( X ) } } } } } } } } } 9 1 9 4 1 2 } 0 8 6 1 3 3 3 4 6 6 1 2 3 , , , , , , 8 8 8 3 3 2 , , , , 3 3 3 3 1 1 1 3 3 5 { { { { { { { { { { { 3 } { 18 } { 33 } { 52 } ∅ Figure: r 10 ( Y ) Dobrinen Higher dimensional Ellentuck spaces University of Denver 16 / 43
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. Dobrinen Higher dimensional Ellentuck spaces University of Denver 17 / 43
( E 2 , ⊆ , r ) is a topological Ramsey space Thm. [D] ( E 2 , ⊆ , r ) is a topological Ramsey space. Dobrinen Higher dimensional Ellentuck spaces University of Denver 18 / 43
( E 2 , ⊆ , r ) is a topological Ramsey space Thm. [D] ( E 2 , ⊆ , r ) is a topological Ramsey space. Thus, every subset of E 2 with the property of Baire is Ramsey. Dobrinen Higher dimensional Ellentuck spaces University of Denver 18 / 43
( E 2 , ⊆ , r ) is a topological Ramsey space Thm. [D] ( E 2 , ⊆ , r ) is a topological Ramsey space. Thus, every subset of E 2 with the property of Baire is Ramsey. Def. A set X ⊆ E 2 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 E 2 . For a ∈ AR and X ∈ E 2 , [ a , X ] := { Y ∈ E 2 : a ❁ Y ⊆ X } . The Ellentuck topology is generated by basic open sets of the form [ a , X ] , where a ∈ AR and X ∈ E 2 . Dobrinen Higher dimensional Ellentuck spaces University of Denver 18 / 43
P ( ω × ω ) / Fin ⊗ 2 is forcing equivalent to a new topological Ramsey space ⊗ 2 ⊗ 2 ( E 2 , ⊆ Fin ) is forcing equivalent to ((Fin ⊗ 2 ) + , ⊆ Fin ). Dobrinen Higher dimensional Ellentuck spaces University of Denver 19 / 43
P ( ω × ω ) / Fin ⊗ 2 is forcing equivalent to a new topological Ramsey space ⊗ 2 ⊗ 2 ( E 2 , ⊆ Fin ) is forcing equivalent to ((Fin ⊗ 2 ) + , ⊆ Fin ). (Below any member A ∈ (Fin ⊗ 2 ) + is some B ⊆ A which is an isomorphic copy of W 2 , and below B , there is a dense subset of (Fin ⊗ 2 ) + ↾ B isomorphic to E 2 .) 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 ⊆ ω 3 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 ⊆ ω 3 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 ( ω 3 ) / Fin ⊗ 3 adds a generic ultrafilter G 3 on ω 3 such that its projection to the first two coordinates is a generic ultrafilter forced by P ( ω 2 ) / 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 ⊆ ω 3 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 ( ω 3 ) / Fin ⊗ 3 adds a generic ultrafilter G 3 on ω 3 such that its projection to the first two coordinates is a generic ultrafilter forced by P ( ω 2 ) / 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 E 3 forcing equivalent ⊗ 3 (when partially ordered by ⊆ Fin ) to P ( ω 3 ) / Fin ⊗ 3 . Dobrinen Higher dimensional Ellentuck spaces University of Denver 20 / 43
The structure behind E 3 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) (0,0,0) (0,0,1) (0,0,2) (0,0,3) (0,1,1) (0,1,2) (0,1,3) (0,2,2) (0,2,3) (0,3,3) (1,1,1) (1,1,2) (1,1,3) (1,2,2) (1,2,3) (1,3,3) (2,2,2) (2,2,3) (2,3,3) (3,3,3) (0 , 0) (0 , 1) (0 , 2) (0 , 3) (1 , 1) (1 , 2) (1 , 3) (2 , 2) (2 , 3) (3 , 3) (0) (1) (2) (3) ∅ Figure: ω � ↓≤ 3 Dobrinen Higher dimensional Ellentuck spaces University of Denver 21 / 43
{ 16,17,18 } { 16,17,28 } { 16,29,30 } { 31,32,33 } { 0,11,12 } { 0,11,21 } { 0,22,23 } { 6,14,15 } { 6,14,25 } { 6,26,27 } { 0,1,19 } { 0,4,10 } { 0,4,20 } { 6,7,13 } { 6,7,24 } { 0,1,2 } { 0,1,3 } { 0,1,9 } { 0,4,5 } { 6,7,8 } { 0 , 1 } { 0 , 4 } { 0 , 11 } { 0 , 22 } { 6 , 7 } { 6 , 14 } { 6 , 26 } { 16 , 17 } { 16 , 29 } { 31 , 32 } { 0 } { 6 } { 16 } { 31 } ∅ Figure: The maximum member of E 3 , W 3 ⊆ [ ω ] 3 Dobrinen Higher dimensional Ellentuck spaces University of Denver 22 / 43
{ 16,17,18 } { 16,17,28 } { 16,29,30 } { 31,32,33 } { 0,11,12 } { 0,11,21 } { 0,22,23 } { 6,14,15 } { 6,14,25 } { 6,26,27 } { 0,1,19 } { 0,4,10 } { 0,4,20 } { 6,7,13 } { 6,7,24 } { 0,1,2 } { 0,1,3 } { 0,1,9 } { 0,4,5 } { 6,7,8 } { 0 , 1 } { 0 , 4 } { 0 , 11 } { 0 , 22 } { 6 , 7 } { 6 , 14 } { 6 , 26 } { 16 , 17 } { 16 , 29 } { 31 , 32 } { 0 } { 6 } { 16 } { 31 } ∅ Figure: The maximum member of E 3 , W 3 ⊆ [ ω ] 3 (0,0,0) (0,0,1) (0,0,2) (0,0,3) (0,1,1) (0,1,2) (0,1,3) (0,2,2) (0,2,3) (0,3,3) (1,1,1) (1,1,2) (1,1,3) (1,2,2) (1,2,3) (1,3,3) (2,2,2) (2,2,3) (2,3,3) (3,3,3) (0 , 0) (0 , 1) (0 , 2) (0 , 3) (1 , 1) (1 , 2) (1 , 3) (2 , 2) (2 , 3) (3 , 3) (0) (1) (2) (3) ∅ Figure: ω � ↓≤ 3 Dobrinen Higher dimensional Ellentuck spaces University of Denver 22 / 43
The space E 3 { 16,17,18 } { 16,17,28 } { 16,29,30 } { 31,32,33 } { 0,11,12 } { 0,11,21 } { 0,22,23 } { 6,14,15 } { 6,14,25 } { 6,26,27 } { 0,1,19 } { 0,4,10 } { 0,4,20 } { 6,7,13 } { 6,7,24 } { 0,1,2 } { 0,1,3 } { 0,1,9 } { 0,4,5 } { 6,7,8 } { 0 , 1 } { 0 , 4 } { 0 , 11 } { 0 , 22 } { 6 , 7 } { 6 , 14 } { 6 , 26 } { 16 , 17 } { 16 , 29 } { 31 , 32 } { 0 } { 6 } { 16 } { 31 } ∅ Figure: W 3 Dobrinen Higher dimensional Ellentuck spaces University of Denver 23 / 43
The space E 3 { 16,17,18 } { 16,17,28 } { 16,29,30 } { 31,32,33 } { 0,11,12 } { 0,11,21 } { 0,22,23 } { 6,14,15 } { 6,14,25 } { 6,26,27 } { 0,1,19 } { 0,4,10 } { 0,4,20 } { 6,7,13 } { 6,7,24 } { 0,1,2 } { 0,1,3 } { 0,1,9 } { 0,4,5 } { 6,7,8 } { 0 , 1 } { 0 , 4 } { 0 , 11 } { 0 , 22 } { 6 , 7 } { 6 , 14 } { 6 , 26 } { 16 , 17 } { 16 , 29 } { 31 , 32 } { 0 } { 6 } { 16 } { 31 } ∅ Figure: W 3 X ∈ E 3 iff X ⊆ W 3 and X ∼ = W 3 as a tree, and also with respect to the ≺ order of the node labels. Dobrinen Higher dimensional Ellentuck spaces University of Denver 23 / 43
The space E 3 { 16,17,18 } { 16,17,28 } { 16,29,30 } { 31,32,33 } { 0,11,12 } { 0,11,21 } { 0,22,23 } { 6,14,15 } { 6,14,25 } { 6,26,27 } { 0,1,19 } { 0,4,10 } { 0,4,20 } { 6,7,13 } { 6,7,24 } { 0,1,2 } { 0,1,3 } { 0,1,9 } { 0,4,5 } { 6,7,8 } { 0 , 1 } { 0 , 4 } { 0 , 11 } { 0 , 22 } { 6 , 7 } { 6 , 14 } { 6 , 26 } { 16 , 17 } { 16 , 29 } { 31 , 32 } { 0 } { 6 } { 16 } { 31 } ∅ Figure: W 3 X ∈ E 3 iff X ⊆ W 3 and X ∼ = W 3 as a tree, and also with respect to the ≺ order of the node labels. Y ≤ X iff Y ⊆ X . Dobrinen Higher dimensional Ellentuck spaces University of Denver 23 / 43
} } } } 3 } 1 5 7 3 } } 4 2 3 3 , 2 2 9 3 , , , 1 1 6 3 , , , 1 1 1 1 1 3 , 1 , , , , , , 0 0 0 0 0 0 3 { { { { { { { { 0 , 1 } { 0 , 11 } { 0 , 36 } { 31 , 32 } { 0 } { 31 } ∅ Figure: r 7 ( Y ), a typical finite approximation to a member of E 3 ) ) ) ) ) ) ) 0 1 2 1 2 2 1 , , , , , , , 0 0 0 1 1 2 1 , , , , , , , 0 0 0 0 0 0 1 ( ( ( ( ( ( ( (0 , 0) (0 , 1) (0 , 2) (1 , 1) (0) (1) () Dobrinen Higher dimensional Ellentuck spaces University of Denver 24 / 43
We now define the spaces E k , k ≥ 2, in general. Dobrinen Higher dimensional Ellentuck spaces University of Denver 25 / 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 . 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 ( j 0 , . . . , j p − 1 ) and ( l 0 , . . . , l q − 1 ) in ω � ↓≤ k with p , q ≥ 1, define ( j 0 , . . . , j p − 1 ) ≺ ( l 0 , . . . , l q − 1 ) if and only if either 1 j p − 1 < l q − 1 , or 2 j p − 1 = l q − 1 and ( j 0 , . . . , j p − 1 ) < lex ( l 0 , . . . , l q − 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 ( j 0 , . . . , j p − 1 ) and ( l 0 , . . . , l q − 1 ) in ω � ↓≤ k with p , q ≥ 1, define ( j 0 , . . . , j p − 1 ) ≺ ( l 0 , . . . , l q − 1 ) if and only if either 1 j p − 1 < l q − 1 , or 2 j p − 1 = l q − 1 and ( j 0 , . . . , j p − 1 ) < lex ( l 0 , . . . , l q − 1 ). Let � j m denote the ≺ − m -th member of ω � ↓≤ k . For � l ∈ ω denote the m such that � l = � l ∈ ω � ↓≤ k , we let m � j m . Dobrinen Higher dimensional Ellentuck spaces University of Denver 26 / 43
The spaces E k , k ≥ 2 � W k is the image of the function � l �→ { m : � j m ⊑ � l } , � l ∈ ω � ↓≤ k . Dobrinen Higher dimensional Ellentuck spaces University of Denver 27 / 43
The spaces E k , k ≥ 2 � W k is the image of the function � l �→ { m : � j m ⊑ � l } , � l ∈ ω � ↓≤ k . We say that � X is an E k -tree if � X is a function from ω � ↓≤ k into � W k such that j m | ∩ � j m ) ∈ [ ω ] | � (i) For each m < ω , � X ( � W k ; (ii) For all 1 ≤ m < ω , max( � X ( � j m )) < max( � X ( � j m +1 )); (iii) For all m , n < ω , � X ( � j m ) ❁ � X ( � j n ) if and only if � j m ❁ � j n . Dobrinen Higher dimensional Ellentuck spaces University of Denver 27 / 43
The spaces E k , k ≥ 2 � W k is the image of the function � l �→ { m : � j m ⊑ � l } , � l ∈ ω � ↓≤ k . We say that � X is an E k -tree if � X is a function from ω � ↓≤ k into � W k such that j m | ∩ � j m ) ∈ [ ω ] | � (i) For each m < ω , � X ( � W k ; (ii) For all 1 ≤ m < ω , max( � X ( � j m )) < max( � X ( � j m +1 )); (iii) For all m , n < ω , � X ( � j m ) ❁ � X ( � j n ) if and only if � j m ❁ � j n . The space E k consists of all X := [ � X ], where � X is an E k -tree. Dobrinen Higher dimensional Ellentuck spaces University of Denver 27 / 43
The spaces E k , k ≥ 2 � W k is the image of the function � l �→ { m : � j m ⊑ � l } , � l ∈ ω � ↓≤ k . We say that � X is an E k -tree if � X is a function from ω � ↓≤ k into � W k such that j m | ∩ � j m ) ∈ [ ω ] | � (i) For each m < ω , � X ( � W k ; (ii) For all 1 ≤ m < ω , max( � X ( � j m )) < max( � X ( � j m +1 )); (iii) For all m , n < ω , � X ( � j m ) ❁ � X ( � j n ) if and only if � j m ❁ � j n . The space E k consists of all X := [ � X ], where � X is an E k -tree. For X , Y ∈ E k , Y ≤ X iff Y ⊆ X . Dobrinen Higher dimensional Ellentuck spaces University of Denver 27 / 43
The spaces E k , k ≥ 2 � W k is the image of the function � l �→ { m : � j m ⊑ � l } , � l ∈ ω � ↓≤ k . We say that � X is an E k -tree if � X is a function from ω � ↓≤ k into � W k such that j m | ∩ � j m ) ∈ [ ω ] | � (i) For each m < ω , � X ( � W k ; (ii) For all 1 ≤ m < ω , max( � X ( � j m )) < max( � X ( � j m +1 )); (iii) For all m , n < ω , � X ( � j m ) ❁ � X ( � j n ) if and only if � j m ❁ � j n . The space E k consists of all X := [ � X ], where � X is an E k -tree. For X , Y ∈ E k , Y ≤ X iff Y ⊆ X . For each n < ω , the n -th finite aproximation r n ( X ) is X ∩ ( { � i p : p < n } × W k ), where ( � i p : p < ω ) is the ≺ -wellordering on ω � ↓ k . Dobrinen Higher dimensional Ellentuck spaces University of Denver 27 / 43
The E k are high dimensional Ellentuck spaces Thm. [D] For each 2 ≤ k < ω , ( E k , ⊆ , r ) is a topological Ramsey space. Dobrinen Higher dimensional Ellentuck spaces University of Denver 28 / 43
The E k are high dimensional Ellentuck spaces Thm. [D] For each 2 ≤ k < ω , ( E k , ⊆ , r ) is a topological Ramsey space. Remarks. 1 Each space E k +1 is comprised of ω many copies of E k . Dobrinen Higher dimensional Ellentuck spaces University of Denver 28 / 43
The E k are high dimensional Ellentuck spaces Thm. [D] For each 2 ≤ k < ω , ( E k , ⊆ , r ) is a topological Ramsey space. Remarks. 1 Each space E k +1 is comprised of ω many copies of E k . 2 Moreover, each projection of E k to levels 1 through j produces a copy of E j . Dobrinen Higher dimensional Ellentuck spaces University of Denver 28 / 43
The E k are high dimensional Ellentuck spaces Thm. [D] For each 2 ≤ k < ω , ( E k , ⊆ , r ) is a topological Ramsey space. Remarks. 1 Each space E k +1 is comprised of ω many copies of E k . 2 Moreover, each projection of E k to levels 1 through j produces a copy of E j . 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.) Dobrinen Higher dimensional Ellentuck spaces University of Denver 28 / 43
Initial Tukey and Rudin-Keisler structures below G k , k ≥ 2 Thm. [D] Let G k denote the generic ultrafilter forced by P ( ω k ) / Fin ⊗ k . Dobrinen Higher dimensional Ellentuck spaces University of Denver 29 / 43
Initial Tukey and Rudin-Keisler structures below G k , k ≥ 2 Thm. [D] Let G k denote the generic ultrafilter forced by P ( ω k ) / Fin ⊗ k . 1 If V ≤ T G k , then V ≡ T π l ( G k ) for some l ≤ k . Dobrinen Higher dimensional Ellentuck spaces University of Denver 29 / 43
Initial Tukey and Rudin-Keisler structures below G k , k ≥ 2 Thm. [D] Let G k denote the generic ultrafilter forced by P ( ω k ) / Fin ⊗ k . 1 If V ≤ T G k , then V ≡ T π l ( G k ) for some l ≤ k . 2 Thus, the Tukey equivalence classes of (nonprincipal) ultrafilters Tukey reducible to G k form a chain of length k . Dobrinen Higher dimensional Ellentuck spaces University of Denver 29 / 43
Initial Tukey and Rudin-Keisler structures below G k , k ≥ 2 Thm. [D] Let G k denote the generic ultrafilter forced by P ( ω k ) / Fin ⊗ k . 1 If V ≤ T G k , then V ≡ T π l ( G k ) for some l ≤ k . 2 Thus, the Tukey equivalence classes of (nonprincipal) ultrafilters Tukey reducible to G k form a chain of length k . 3 Further, the Rudin-Keisler equivalence classes of (nonprincipal) ultrafilters RK-reducible to G k form a chain of length k . Dobrinen Higher dimensional Ellentuck spaces University of Denver 29 / 43
Initial Tukey and Rudin-Keisler structures below G k , k ≥ 2 Thm. [D] Let G k denote the generic ultrafilter forced by P ( ω k ) / Fin ⊗ k . 1 If V ≤ T G k , then V ≡ T π l ( G k ) for some l ≤ k . 2 Thus, the Tukey equivalence classes of (nonprincipal) ultrafilters Tukey reducible to G k form a chain of length k . 3 Further, the Rudin-Keisler equivalence classes of (nonprincipal) ultrafilters RK-reducible to G k form a chain of length k . Remark. The fact that E k is dense below any member of ( Fin ⊗ k ) + provides a simple way of reading off the partition relations for the generic ultrafilter. Dobrinen Higher dimensional Ellentuck spaces University of Denver 29 / 43
Outline of Proof ⊗ k 1 Show ( E k , ⊆ Fin ) is forcing equivalent to P ( ω k ) / Fin ⊗ k . Dobrinen Higher dimensional Ellentuck spaces University of Denver 30 / 43
Outline of Proof ⊗ k 1 Show ( E k , ⊆ Fin ) is forcing equivalent to P ( ω k ) / Fin ⊗ k . 2 Prove ( E k , ≤ , r ) is a topological Ramsey space. Dobrinen Higher dimensional Ellentuck spaces University of Denver 30 / 43
Outline of Proof ⊗ k 1 Show ( E k , ⊆ Fin ) is forcing equivalent to P ( ω k ) / Fin ⊗ k . 2 Prove ( E k , ≤ , r ) is a topological Ramsey space. 3 Prove a Ramsey-classification theorem for equivalence relations on fronts on E k , extending the Pudl´ ak-R¨ odl Theorem for the Ellentuck space. Dobrinen Higher dimensional Ellentuck spaces University of Denver 30 / 43
Outline of Proof ⊗ k 1 Show ( E k , ⊆ Fin ) is forcing equivalent to P ( ω k ) / Fin ⊗ k . 2 Prove ( E k , ≤ , r ) is a topological Ramsey space. 3 Prove a Ramsey-classification theorem for equivalence relations on fronts on E k , extending the Pudl´ ak-R¨ odl 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 ⊗ k 1 Show ( E k , ⊆ Fin ) is forcing equivalent to P ( ω k ) / Fin ⊗ k . 2 Prove ( E k , ≤ , r ) is a topological Ramsey space. 3 Prove a Ramsey-classification theorem for equivalence relations on fronts on E k , extending the Pudl´ ak-R¨ odl 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 G k , apply Basic Cofinal Maps Theorem to find a front F on E k and an f : F → ω such that V = f ( �G k |F� ). Dobrinen Higher dimensional Ellentuck spaces University of Denver 30 / 43
Outline of Proof ⊗ k 1 Show ( E k , ⊆ Fin ) is forcing equivalent to P ( ω k ) / Fin ⊗ k . 2 Prove ( E k , ≤ , r ) is a topological Ramsey space. 3 Prove a Ramsey-classification theorem for equivalence relations on fronts on E k , extending the Pudl´ ak-R¨ odl 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 G k , apply Basic Cofinal Maps Theorem to find a front F on E k and an f : F → ω such that V = f ( �G k |F� ). 6 Apply the Ramsey-classification theorem for equivalence relations on fronts and analyze f ( �G k |F� ). Dobrinen Higher dimensional Ellentuck spaces University of Denver 30 / 43
Def. A family of finite approximations F is a front on E k iff (i) ∀ X ∈ E k , ∃ 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 E k iff (i) ∀ X ∈ E k , ∃ 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 E k and f : F → ω . Then there exists an X ∈ E k 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 ) . Dobrinen Higher dimensional Ellentuck spaces University of Denver 32 / 43
Ramsey-classification Theorem for equivalence relations on fronts Thm. [D] Let F be a front on E k and f : F → ω . Then there exists an X ∈ E k 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¨ odl 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 . Dobrinen Higher dimensional Ellentuck spaces University of Denver 32 / 43
Ramsey-classification Theorem for equivalence relations on fronts Thm. [D] Let F be a front on E k and f : F → ω . Then there exists an X ∈ E k 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¨ odl 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 E k . Suppose ϕ and ϕ ′ are irreducible maps canonizing R . Then there is an A ∈ E k 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 AR n , the canonical equivalence relations are given by projection maps of the form ϕ ( a (0) , . . . , a ( n − 1)) = ( π j 0 ( a (0)) , . . . , π j n − 1 ( a ( n − 1))) , where π j ( a ( i )) is the projection of a ( i ) to its first j levels (in the tree W k ). Dobrinen Higher dimensional Ellentuck spaces University of Denver 33 / 43
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