A new class of Ramsey-Classification Theorems and their Applications in the Tukey Theory of Ultrafilters, Parts 1 and 2

Motivated by Tukey classification problems, we develop a new hierarchy of topological Ramsey spaces R α , α < ω 1 . These spaces form a natural hierarchy of complexity, R 0 being the Ellentuck space [Erik Ellentuck, A new proof that analytic sets are Ramsey, Journal of Symbolic Logic 39 (1974), 163–165], and for each α < ω 1 , R α + 1 coming immediately after R α in complexity. Associated with each R α is an ultrafilter U α , which is Ramsey for R α , 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 , 1 ⩽ α < ω 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 , for each 1 ⩽ α < ω 1 : Every nonprincipal ultrafilter which is Tukey reducible to U α 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 form a descending chain of rapid p-points of order type α + 1 .