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

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.