A new class of Ramsey-classification theorems and their applications in the Tukey theory of ultrafilters, Part 2

Abstract

Motivated by Tukey classification problems and building on work in Part 1, we develop a new hierarchy of topological Ramsey spaces R α , α < ω 1 . These spaces form a natural hierarchy of complexity, R 0 being the Ellentuck space, and for each α < ω 1 , R α+1 coming immediately after R α in complexity. Associated with each R α is an ultrafilter U α , which is Ramsey for R α , and in particular, is a rapid p-point satisfying certain partition properties. We prove Ramsey-classification theorems for equivalence relations on fronts on R α , 2 ≤ α < ω 1 . These form a hierarchy of extensions of the Pudlak-Rödl Theorem canonizing equivalence relations on barriers on the Ellentuck space. We then apply our Ramsey-classification theorems to completely classify all Rudin-Keisler equivalence classes of ultrafilters which are Tukey reducible to U α , for each 2 ≤ α < ω 1 : Every nonprincipal ultrafilter which is Tukey reducible to U α is isomorphic to a countable iteration of Fubini products of ultrafilters from among a fixed countable collection of rapid p-points. Moreover, we show that the Tukey types of nonprincipal ultrafilters Tukey reducible to U α form a descending chain of rapid p-points of order type α + 1.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 4628 NATASHA DOBRINEN AND STEVO TODORCEVIC equivalence relations for uniform barriers on R 1 of rank n. In this paper, we prove similar results for all 2 ≤ α < ω 1 . In Theorem 4.22, we show that for all 2 ≤ α < ω 1 , for any uniform barrier B on R α of finite rank and any equivalence relation E on B, there is an X ∈ R α such that E restricted to the members of B coming from within X is exactly one of the canonical equivalence relations. For finite α, there are finitely many canonical equivalence relations on uniform barriers of finite rank; these are represented by a certain collection of finite trees. Moreover, the numbers of canonical equivalence relations for finite α are given by a recursive function. For infinite α, there are infinitely many canonical equivalence relations on uniform barriers of finite rank, represented by tree-like structures. These theorems generalize the Erdős-Rado Theorem [8] for uniform barriers of finite rank on the Ellentuck space, namely, those of the form [N] n .In the main theorem of this paper, Theorem 4.12, we prove new Ramsey-classification theorems for all barriers (and moreover all fronts) on the topological Ramsey spaces R α , 2 ≤ α < ω 1 . We prove that for any barrier B on R α and any equivalence relation on B, there is an inner Sperner map which canonizes the equivalence relation. This generalizes our analogous theorem for R 1 in [5], which in turn generalized the Pudlak-Rödl Theorem [13] for barriers on the Ellentuck space. These classification theorems were motivated by the following.Recently the second author (see Theorem 24 in [14]) has made a connection between the Ramsey-classification theory (also known as the canonical Ramsey theory) and the Tukey classificati...

Infinite-dimensional Ellentuck spaces and Ramsey-classification theorems

“…The proof of Lemma 40 is straightforward, using the Fusion Lemma 24; being very similar to that of Lemma 4.6 in [12], we omit it.…”

“…Building on prior work of Carlson and Simpson, Todorcevic distilled four axioms which, when satisfied, guarantee that a space is a toplogical Ramsey space (see Section 3). This axiomatic approach to topological Ramsey spaces paved the way for recent work involving connections between ultrafilters satisfying partition properties and constructions of new topological Ramsey spaces (see [12], [13], [14], and [1]).…”

“…In [20], Todorcevic proved that the Tukey type of any Ramsey ultrafilter is minimal via a judicious application of the Pudlák-Rödl Theorem on the Ellentuck space. In [12] and [13], Dobrinen and Todorcevic constructed a new class of topological Ramsey spaces which are dense in the partial orderings of Laflamme in [23] for ultrafilters with weaker partition properties. By proving and applying Ramsey-classification theorems, they found the exact initial Rudin-Keisler and Tukey structures for this class of ultrafilters.…”

Creature forcing and topological Ramsey spaces

“…The first two tutorials focused on areas (1) -(3). We presented work from [10], [11] [9], [3], [4], and [2], in which dense subsets of forcings generating ultrafilters satisfying some weak partition properties were shown to form topological Ramsey spaces. Having obtained the canonical equivalence relations on fronts for these topological Ramsey spaces, they may be applied to obtain the precise initial Rudin-Keisler and Tukey structures.…”

Continuous and other finitely generated canonical cofinal maps on ultrafilters

“…A map f : U → V is monotone if for any X, Y ∈ U, X ⊇ Y implies f (X) ⊇ f (Y ). It is not hard to show that whenever U ≥ T V, then there is a monotone cofinal map witnessing this (see Fact 6 of [8]).…”