AbstractView references

Motivated by a Tukey classification problem, we develop a new topological Ramsey space R₁ that in its complexity comes immediately after the classical Ellentuck space. Associated with R₁ is an ultrafilter U₁ which is weakly Ramsey but not Ramsey. We prove a canonization theorem for equivalence relations on fronts on R₁. This is analogous to the Pudlak-Rödl Theorem canonizing equivalence relations on barriers on the Ellentuck space. We then apply our canonization theorem to completely classify all Rudin-Keisler equivalence classes of ultrafilters which are Tukey reducible to U₁: Every ultrafilter which is Tukey reducible to U₁ is isomorphic to a countable iteration of Fubini products of ultrafilters from among a fixed countable collection of ultrafilters. Moreover, we show that there is exactly one Tukey type of nonprincipal ultrafilters strictly below that of U₁, namely the Tukey type of a Ramsey ultrafilter. © 2013 American Mathematical Society.