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 $\mathcal {R}_1$ that in its complexity comes immediately after the classical Ellentuck space. Associated with $\mathcal {R}_1$ is an ultrafilter $\mathcal {U}_1$ which is weakly Ramsey but not Ramsey. We prove a canonization theorem for equivalence relations on fronts on $\mathcal {R}_1$. 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 $\mathcal {U}_1$: Every ultrafilter which is Tukey reducible to $\mathcal {U}_1$ 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 $\mathcal {U}_1$, namely the Tukey type of a Ramsey ultrafilter.