A new class of Ramsey-classification theorems and their applications in the Tukey theory of ultrafilters

Motivated by Tukey classification problems and building on work in \cite{Dobrinen/Todorcevic11}, we develop a new hierarchy of topological Ramsey spaces ${\mathcal{R}}_{\alpha }$, $\alpha <{\omega }_1$. These spaces form a natural hierarchy of complexity, ${\mathcal{R}}_0$ being the Ellentuck space, and for each $\alpha <{\omega }_1$, ${\mathcal{R}}_{\alpha +1}$ coming immediately after ${\mathcal{R}}_{\alpha }$ in complexity. Associated with each ${\mathcal{R}}_{\alpha }$ is an ultrafilter ${\mathcal{U}}_{\alpha }$, which is Ramsey for ${\mathcal{R}}_{\alpha }$, and in particular, is a rapid p-point satisfying certain partition properties. We prove Ramsey-classification theorems for equivalence relations on fronts on ${\mathcal{R}}_{\alpha }$, $2\le \alpha <{\omega }_1$. These are analogous to the Pudlak-\Rodl\ Theorem canonizing equivalence relations on barriers on the Ellentuck space. We then apply our Ramsey-classification theorems to completely classify all Rudin-Keisler equivalence classes of ultrafilters which are Tukey reducible to ${\mathcal{U}}_{\alpha }$, for each $2\le \alpha <{\omega }_1$: Every ultrafilter which is Tukey reducible to ${\mathcal{U}}_{\alpha }$ is isomorphic to a countable iteration of Fubini products of ultrafilters from among a fixed countable collection of rapid p-points. Moreover, we show that the Tukey types of nonprincipal ultrafilters Tukey reducible to ${\mathcal{U}}_{\alpha }$ form a descending chain of order type $\alpha +1$.