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<inf>α</inf>, α < ω<inf>1</inf>. These spaces form a natural hierarchy of complexity, R<inf>0</inf> being the Ellentuck space, and for each α < ω<inf>1</inf>, R<inf>α+1</inf> coming immediately after R<inf>α</inf> in complexity. Associated with each R<inf>α</inf> is an ultrafilter U<inf>α</inf>, which is Ramsey for R<inf>α</inf>, and in particular, is a rapid p-point satisfying certain partition properties. We prove Ramsey-classification theorems for equivalence relations on fronts on R<inf>α</inf>, 2 ≤ α < ω<inf>1</inf>. 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<inf>α</inf>, for each 2 ≤ α < ω<inf>1</inf>: Every nonprincipal ultrafilter which is Tukey reducible to U<inf>α</inf> 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<inf>α</inf> form a descending chain of rapid p-points of order type α + 1.