A Ramsey-Classification Theorem and its Application in the Tukey Theory of Ultrafilters

Abstract: Motivated by a Tukey classification problem we develop here a new topological Ramsey space $\mathcal{R}_1$ that in its complexity comes immediately after the classical is a natural Ellentuck space \cite{MR0349393}. 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-\Rodl\ 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 a Ramsey ultrafilter.