A New Class of Ramsey-classification Theorems and Their Application in the Tukey Theory of Ultrafilters, Part 1

Motivated by a Tukey classification problem we develop here a new topological Ramsey space R1 that in its complexity comes immediately after the classical Ellentuck space [8]. 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 extends 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.