Higher dimensional Ellentuck spaces

Transcription

1 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

2 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. E k (2 k < ω) spaces appear in Work in this talk [Dobrinen] High dimensional Ellentuck spaces and initial chains in the Tukey types of non p-points, Journal of Symbolic Logic, to appear. E B 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, Dobrinen Higher dimensional Ellentuck spaces University of Denver 2 / 43

3 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

4 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

5 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 G 2 on base set ω ω. G 2 is neither a p-point, nor a Fubini product of p-points, but the projection to the first coordinates π 1 (G 2 ) is a Ramsey ultrafilter. Dobrinen Higher dimensional Ellentuck spaces University of Denver 5 / 43

6 In [Blass/Dobrinen/Raghavan], we showed the following: Thm. In V [G 2 ], G 2 (B) is a weak p-point; (B) has the best partition property G 2 (G 2 ) 2 k,3 have; (D,R) is not Tukey maximum; (D) (G 2, ) T ([ω 1 ] <ω, ); (D) G 2 > T π 1 (G 2 ); (R) is not basically generated. a non-p-point can This left open what exactly is Tukey reducible to G 2 ; i.e. What is the initial Tukey structure below G 2. Dobrinen Higher dimensional Ellentuck spaces University of Denver 6 / 43

7 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 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

8 Review Dobrinen Higher dimensional Ellentuck spaces University of Denver 8 / 43

9 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

10 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

11 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

12 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. (0,0) (0,1) (0,2) (0,3) (0,4) (1,1) (1,2) (1,3) (1,4) (2,2) (2,3) (2,4) (3,3) (3,4) (4,4) (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

13 Constructing the maximal member of E {7} (0,0) (0,1) (0,2) (0,3) (0,4) (1,1) (1,2) (1,3) (1,4) (2,2) (2,3) (2,4) (3,3) (3,4) (4,4) (0) (1) (2) (3) (4) () () (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

14 {0,1} {0,2} {0,5} {0,9} {0,14} {3,4} {3,6} {3,10} {3,15} {7,8} {7,11} {7,16} {12,13} {12,17} {18,19} {0} {3} {7} {12} {18} Figure: Maximum element W 2 [ω] 2 of E 2 (0,0) (0,1) (0,2) (0,3) (0,4) (1,1) (1,2) (1,3) (1,4) (2,2) (2,3) (2,4) (3,3) (3,4) (4,4) (0) (1) (2) (3) (4) () Figure: ω () (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

15 The space E 2 {0,1} {0,2} {0,5} {0,9} {0,14} {3,4} {3,6} {3,10} {3,15} {7,8} {7,11} {7,16} {12,13} {12,17} {18,19} {0} {3} {7} {12} {18} Figure: W 2 [ω] 2 X E 2 iff X is a subset of W 2 such that (1) ˆX is tree-isomorphic to Ŵ2, and (2) max values of the nodes of ˆX 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

16 Typical finite approximations to members of E 2 {0,1} {0,5} {0,14} {0,35} {7,11} {7,16} {18,19} {0} {7} {18} Figure: r 7 (X ) {3,6} {3,10} {3,28} {3,36} {18,19} {18,31} {18,39} {33,34} {33,41} {52,62} {3} {18} {33} {52} Figure: r 10 (Y ) Dobrinen Higher dimensional Ellentuck spaces University of Denver 16 / 43

17 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

18 (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

19 P(ω ω)/fin 2 is forcing equivalent to a new topological Ramsey space (E 2, 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 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

20 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 (when partially ordered by Fin 3 ) to P(ω 3 )/Fin 3. Dobrinen Higher dimensional Ellentuck spaces University of Denver 20 / 43

21 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

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

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

24 {0,1,2} {0,1,9} {0,1,34} {0,11,21} {0,11,35} {0,36,37} {31,32,33} {0, 1} {0, 11} {0, 36} {31, 32} {0} {31} Figure: r 7 (Y ), a typical finite approximation to a member of E 3 (0,0,0) (0,0,1) (0,0,2) (0,1,1) (0,1,2) (0,2,2) (1,1,1) (0, 0) (0, 1) (0, 2) (1, 1) (0) (1) Dobrinen Higher dimensional Ellentuck spaces University of Denver 24 / 43 ()

25 We now define the spaces E k, k 2, in general. Dobrinen Higher dimensional Ellentuck spaces University of Denver 25 / 43

26 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 k, we let m l ω denote the m such that l = j m. Dobrinen Higher dimensional Ellentuck spaces University of Denver 26 / 43

27 The spaces E k, k 2 Ŵ 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 Ŵk such that (i) For each m < ω, X ( j m ) [ω] j m Ŵk; (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

28 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

29 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

30 Outline of Proof 1 Show (E k, Fin k ) 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 Pudlák-Rödl 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

31 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 â. 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

32 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 Pudlák-Rödl 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

33 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 j0 (a(0)),..., π jn 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

34 Basic Cofinal Maps from G k Def. Given Y B k := G k E k, a monotone map g : B k Y P(ω) is basic if there is a map ĝ : AR Y [ω] <ω such that 1 (monotonicity) For all s, t AR Y, s t ĝ(s) ĝ(t); 2 (initial segment preserving) For s t in AR Y, ĝ(s) ĝ(t); 3 (ĝ represents g) For each V B k Y, g(v ) = n<ω ĝ(r n(v )). Thm. (Basic monotone maps on G k ) [D] Let G k generic for P(ω k )/Fin k. In V [G k ], for each monotone function g : G k P(ω), there is a Y B k such that g (B k Y ) is basic. Dobrinen Higher dimensional Ellentuck spaces University of Denver 34 / 43

35 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). Dobrinen Higher dimensional Ellentuck spaces University of Denver 35 / 43

36 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 E B. Such spaces E B 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. Dobrinen Higher dimensional Ellentuck spaces University of Denver 36 / 43

37 The first infinite dimensional Ellentuck space Let S denote {a [ω] <ω : a = min(a) + 1}. S is the Schreier barrier. Fin S is an ideal on S: X S is in Fin S iff for all but finitely many k < ω, {a \ {k} : a X and min(a) = k} Fin k. X S is in (Fin S ) + iff there are infinitely many k such that {a \ {k} : a X and min(a) = k} (Fin k ) +. P(S)/Fin S is forcing equivalent to ((Fin S ) +, FinS ). Dobrinen Higher dimensional Ellentuck spaces University of Denver 37 / 43

38 We use the form of S to make our template structure of finite non-decreasing sequences of natural numbers. (3,3,3,3) (3,3,3,4) (3,3,4,4) (3,4,4,4) (2,2,2) (2,2,3) (2,2,4) (2,3,3) (2,3,4) (2,4,4) (3, 3, 3) (3, 3, 4) (3, 4, 4) (1, 1) (1, 2) (1, 3) (1, 4) (2, 2) (2, 3) (2, 4) (3, 3) (3, 4) (0) (1) (2) (3) 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

39 Figure: W S (3,3,3,3) (3,3,3,4) (3,3,4,4) (3,4,4,4) (2,2,2) (2,2,3) (2,2,4) (2,3,3) (2,3,4) (2,4,4) (3, 3, 3) (3, 3, 4) (3, 4, 4) (1, 1) (1, 2) (1, 3) (1, 4) (2, 2) (2, 3) (2, 4) (3, 3) (3, 4) (0) (1) (2) (3) Figure: ω S Dobrinen Higher dimensional Ellentuck spaces University of Denver 39 / 43

40 {4,5,6} {4,5,8} {4,5,16} {4,9,10} {4,9,17} {4,18,19} {11,12,13,14} {11, 12, 13} {11,12,13,20} {11,12,21,22} {11, 12, 21} {11,23,24,25} {11, 23, 24} {1, 2} {1, 3} {1, 7} {1, 15} {4, 5} {4, 9} {4, 18} {11, 12} {11, 23} {0} {1} {4} {11} Figure: W S (3,3,3,3) (3,3,3,4) (3,3,4,4) (3,4,4,4) (2,2,2) (2,2,3) (2,2,4) (2,3,3) (2,3,4) (2,4,4) (3, 3, 3) (3, 3, 4) (3, 4, 4) (1, 1) (1, 2) (1, 3) (1, 4) (2, 2) (2, 3) (2, 4) (3, 3) (3, 4) (0) (1) (2) (3) Figure: ω S Dobrinen Higher dimensional Ellentuck spaces University of Denver 40 / 43

41 The space E S, for S the Schreier barrier E S is the collection of all X W S such that 1 for infinitely many k, {a \ {k} : a X and min a = k} E k, 2 if {a X : min a = k} E k, then it is empty, 3 The values of the nodes in ˆX follow the order. 4 Finitization is recursively induced by the finitizations on the E k. {4,5,6} {4,5,8} {4,5,16} {4,9,10} {4,9,17} {4,18,19} {11,12,13,14} {11, 12, 13} {11,12,13,20} {11,12,21,22} {11, 12, 21} {11,23,24,25} {11, 23, 24} {1, 2} {1, 3} {1, 7} {1, 15} {4, 5} {4, 9} {4, 18} {11, 12} {11, 23} {0} {1} {4} {11} Figure: W S Dobrinen Higher dimensional Ellentuck spaces University of Denver 41 / 43

42 Current work Let B be any uniform barrier on ω. Thm. [D] The space E B is a topological Ramsey space. Forcing with (E B, FinB ) is equivalent to forcing with P(B)/Fin B. Thm. [D] 1 Equivalence relations on AR 1 are canonized as uniform fronts on W B ; that is, projections which have the form of a uniform front. 2 The initial Rudin-Keisler structure below the generic ultrafilter G B is the linear ordering of the G B -equivalence classes of the uniform fronts on W B. 3 Special Case: For the Schreier barrier S, the initial Rudin-Keisler structure below the generic ultrafilter G S is the ultrapower of N modulo the projected Ramsey ultrafilter π 1 (G S ). Dobrinen Higher dimensional Ellentuck spaces University of Denver 42 / 43

43 Thm. [D] 1 We have Ramsey-classification theorems canonizing equivalence relations on barriers on E B in terms of irreducible functions. 2 The initial Tukey structure below G B has cardinality c, and contains the linear order of the G B equivalence classes of the uniform fronts on W B. 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