Generalized Ellentuck Spaces and Initial Chains in the Tukey Structure of Non-p-points

In [1], it was shown that the generic ultrafilter G2 forced by P(ω × ω)/(Fin ⊗ Fin) is neither maximum nor mimumum in the Tukey order of ultrafilters, but left open where exactly in the Tukey order it lies. We prove that G2 is in fact Tukey minimal over its projected Ramsey ultrafilter. Furthermore, we prove that for each k ≥ 2, the collection of all nonprincipal ultrafilters Tukey reducible to the generic Gk ultrafilter forced by P(ωk)/Fin⊗k forms a chain of length k. Essential to the proof is the extraction of a dense subset Ek from (Fin⊗k)+ which we prove to be a topological Ramsey space. New Ramsey-classification theorems for equivalence relations on fronts on Ek are proved, extending the Pudlák-Rödl Theorem for fronts on the Ellentuck space, which are applied to find the Tukey structure below Gk. The spaces Ek, k ≥ 2, form a hiearchy of natural generalizations of the Ellentuck space.