A new class of Ramsey-Classification Theorems and their Applications in the Tukey Theory of Ultrafilters, Parts 1 and 2

Motivated by Tukey classification problems and building on work in [4], 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 [6], and for each α < ω1, Rα+1 coming immediately afterRα 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 are analogous to the PudlakRödl 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 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 order type α+ 1.