Authors: Dobrinen, Natasha
Todorčević, Stevo 
Title: A new class of Ramsey-classification theorems and their applications in the Tukey theory of ultrafilters, part 2
Journal: Transactions of the American Mathematical Society
Volume: 367
Issue: 7
First page: 4627
Last page: 4659
Issue Date: 1-Jan-2015
Rank: M21
ISSN: 0002-9947
DOI: 10.1090/S0002-9947-2014-06122-9
Motivated by Tukey classification problems and building on work in Part 1, we develop a new hierarchy of topological Ramsey spaces Rα, α < ω1. These spaces form a natural hierarchy of complexity, R0 being the Ellentuck space, and for each α < ω1, Rα+1 coming immediately after Rα in complexity. Associated with each Rα is an ultrafilter Uα, which is Ramsey for Rα, and in particular, is a rapid p-point satisfying certain partition properties. We prove Ramsey-classification theorems for equivalence relations on fronts on Rα, 2 ≤ α < ω1. 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 Uα, for each 2 ≤ α < ω1: Every nonprincipal ultrafilter which is Tukey reducible to Uα 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 Uα form a descending chain of rapid p-points of order type α + 1.
Publisher: American Mathematical Society

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