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Motivated by Tukey classification problems and building on work in\cite{Dobrinen/Todorcevic11}, we develop a new hierarchy of topological Ramseyspaces $\mathcal{R}_{\alpha}$, $\alpha<\omega_1$. These spaces form a naturalhierarchy of complexity, $\mathcal{R}_0$ being the Ellentuck space, and foreach $\alpha<\omega_1$, $\mathcal{R}_{\alpha+1}$ coming immediately after$\mathcal{R}_{\alpha}$ in complexity. Associated with each$\mathcal{R}_{\alpha}$ is an ultrafilter $\mathcal{U}_{\alpha}$, which isRamsey for $\mathcal{R}_{\alpha}$, and in particular, is a rapid p-pointsatisfying certain partition properties. We prove Ramsey-classificationtheorems for equivalence relations on fronts on $\mathcal{R}_{\alpha}$,$2\le\alpha<\omega_1$. These are analogous to the Pudlak-\Rodl\ Theoremcanonizing equivalence relations on barriers on the Ellentuck space. We thenapply our Ramsey-classification theorems to completely classify allRudin-Keisler equivalence classes of ultrafilters which are Tukey reducible to$\mathcal{U}_{\alpha}$, for each $2\le\alpha<\omega_1$: Every ultrafilter whichis Tukey reducible to $\mathcal{U}_{\alpha}$ is isomorphic to a countableiteration of Fubini products of ultrafilters from among a fixed countablecollection of rapid p-points. Moreover, we show that the Tukey types ofnonprincipal ultrafilters Tukey reducible to $\mathcal{U}_{\alpha}$ form adescending chain of order type $\alpha+1$.

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