A new class of Ramsey-classification theorems and their applications in the Tukey theory of ultrafilters, Part 2

Abstract:

Motivated by Tukey classification problems and building on work in Part 1, 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 form a hierarchy of extensions of the Pudlak-Rödl 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 nonprincipal 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 rapid p-points of order type $\alpha +1$.