AbstractView references

Motivated by Tukey classification problems and building on work in Part 1, we develop a new hierarchy of topological Ramsey spaces R_α, α < ω₁. These spaces form a natural hierarchy of complexity, R₀ being the Ellentuck space, and for each α < ω₁, 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, 2 ≤ α < ω₁. 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 2 ≤ α < ω₁: 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. © 2015 American Mathematical Society.