Abstract

The generic ultrafilter${\cal G}_2 $forced by${\cal P}\left/\left$was recently proved to be neither maximum nor minimum in the Tukey order of ultrafilters, but it was left open where exactly in the Tukey order it lies. We prove${\cal G}_2 $that is in fact Tukey minimal over its projected Ramsey ultrafilter. Furthermore, we prove that for each${\cal G}_2 $, the collection of all nonprincipal ultrafilters Tukey reducible to the generic ultrafilter${\cal G}_k $forced by${\cal P}\left/{\rm{Fin}}^{ \otimes k} $forms a chain of lengthk. Essential to the proof is the extraction of a dense subsetεkfrom +which we prove to be a topological Ramsey space. The spacesεk,k≥ 2, form a hierarchy of high dimensional Ellentuck spaces. New Ramsey-classification theorems for equivalence relations on fronts on εkare proved, extending the Pudlák–Rödl Theorem for fronts on the Ellentuck space, which are applied to find the Tukey and Rudin–Keisler structures below${\cal G}_k $.