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 R α \mathcal {R}_{\alpha } , α > ω 1 \alpha >\omega _1 . These spaces form a natural hierarchy of complexity, R 0 \mathcal {R}_0 being the Ellentuck space, and for each α > ω 1 \alpha >\omega _1 , R α + 1 \mathcal {R}_{\alpha +1} coming immediately after R α \mathcal {R}_{\alpha } in complexity. Associated with each R α \mathcal {R}_{\alpha } is an ultrafilter U α \mathcal {U}_{\alpha } , which is Ramsey for R α \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 R α \mathcal {R}_{\alpha } , 2 ≤ α > ω 1 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 U α \mathcal {U}_{\alpha } , for each 2 ≤ α > ω 1 2\le \alpha >\omega _1 : Every nonprincipal ultrafilter which is Tukey reducible to U α \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 U α \mathcal {U}_{\alpha } form a descending chain of rapid p-points of order type α + 1 \alpha +1 .