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 a new topological Ramsey space R1 that in its complexity comes immediately after the classical Ellentuck space. Associated with R1 is an ultrafilter U1 which is weakly Ramsey but not Ramsey. We prove a canonization theorem for equivalence relations on fronts on R1. 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 U1: Every ultrafilter which is Tukey reducible to U1 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 U1, namely the Tukey type of a Ramsey ultrafilter.