A new class of Ramsey-classification theorems and their application in the Tukey theory of ultrafilters, Part 1

Abstract

Motivated by a Tukey classification problem we develop here a new topological Ramsey space R 1 that in its complexity comes immediately after the classical Ellentuck space [5]. Associated with R 1 is an ultrafilter U 1 which is weakly Ramsey but not Ramsey. We prove a canonization theorem for equivalence relations on fronts on R 1 . This is analogous to the Pudlak-Rödl Theorem canonizing equivalence relations on barriers on the Ellentuck space. We then apply our canonization theorem to completely classify all Rudin-Keisler equivalence classes of ultrafilters which are Tukey reducible to U 1 : Every ultrafilter which is Tukey reducible to U 1 is isomorphic to a countable iteration of Fubini products of ultrafilters from among a fixed countable collection of ultrafilters. Moreover, we show that there is exactly one Tukey type of nonprincipal ultrafilters strictly below that of U 1 , namely the Tukey type of a Ramsey ultrafilter. 1 2 NATASHA DOBRINEN AND STEVO TODORCEVICRecently the second author (see Theorem 24 in [15]) has made a connection between the Ramsey-classification theory (also known as the canonical Ramsey theory) and the Tukey classification theory of ultrafilters on ω. More precisely, he showed that selective ultrafilters realize minimal Tukey types in the class of all ultrafilters on ω by applying the Pudlak-Rödl Ramsey classification result to a given cofinal map from a selective ultrafilter into any other ultrafilter on ω, a map which, on the basis of our previous paper [4], he could assume to be continuous. Recall that the notion of a selective ultrafilter is closely tied to the Ellentuck space on the family of all infinite subsets of ω, or rather the one-dimensional version of the pigeon-hole principle on which the Ellentuck space is based, the principle stating that an arbitrary f : ω → ω is either constant or is one-to-one on an infinite subset of ω. Thus an ultrafilter U on ω is selective if for every map f : ω → ω there is an X ∈ U such that f is either constant or one-to-one on U. Since essentially any other topological Ramsey space has it own notion of a selective ultrafilter living on the set of its 1-approximations (see [12]), the argument for Theorem 24 in [15] is so general that it will give analogous Tukey-classification results for all ultrafilters of this sort provided, of course, that we have the analogues of the Pudlak-Rödl Ramsey-classification result for the corresponding topological Ramsey spaces.This paper is our first step towards a research in this direction. In particular, inspired by work of Laflamme [11], we build a topological Ramsey space R 1 , so that the ultrafilter associated with R 1 is isomorphic to the ultrafilter U 1 forced by Laflamme. In [11], Laflamme forced an ultrafilter, U 1 , which is weakly Ramsey but not Ramsey, and satisfies additional partition properties. Moreover, he showed that U 1 has complete combinatorics over the Solovay model. By work of Blass in [2], U 1 has only one non-trivial Rudin-Keisler equivalence class of ultrafilters strictly below it, ...

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

“…Motivated by a Tukey classification problem and inspired by work of Laflamme in [12] and the second author in [15], we build new topological Ramsey spaces R α , 2 ≤ α < ω 1 . The space R 0 denotes the classical Ellentuck space; the space R 1 was built in [4]. The topological Ramsey spaces R α , α < ω 1 , form a natural hierarchy in terms of complexity.…”

“…Since, as it is well-known, assuming large cardinals, the theory of L(R) cannot be changed by forcing, this gives another perspective to the notion of 'complete combinatorics' of Blass and Laflamme. This paper was motivated by the same two lines of motivation as in [4]. One line of motivation was to find the structure of the Tukey types of nonprincipal ultrafilters Tukey reducible to U α , for all 1 ≤ α < ω 1 .…”

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Motivated by Tukey classification problems and building on work in [4], 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 [6], and for each α < ω 1 , R α+1 coming immediately after R α in complexity.

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“…A NEW CLASS OF RAMSEY-CLASSIFICATION THEOREMS AND THEIR APPLICATIONS IN THE TUKEY THEORY OF ULTRAFILTERS NATASHA DOBRINEN AND STEVO TODORCEVIC Abstract. Motivated by Tukey classification problems and building on work in [4], 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 [6], and for each α < ω 1 , R α+1 coming immediately after R α in complexity.…”

The Tukey Order and Subsets of ω 1

“…The Tukey relation distinguishes between cofinal structures of posets and Tukey equivalence classes are sometimes called cofinality types. Introduced to study Moore-Smith convergence in topology [18,24], Tukey quotients and equivalence are fundamental notions of order theory, and are being actively investigated, especially in connection with partial orders arising naturally in analysis and topology [6,7,8,9,10,15,17,19,20,21,22].…”