A Ramsey-Classification Theorem and its Application in the Tukey Theory of Ultrafilters

TL;DR

This paper introduces a new topological Ramsey space and ultrafilter, providing a classification of ultrafilters Tukey reducible to it, revealing a unique Tukey type below it.

Contribution

It develops a novel topological Ramsey space and applies a canonization theorem to classify ultrafilters Tukey reducible to a specific ultrafilter, extending the Tukey theory of ultrafilters.

Findings

The ultrafilter $_1$ is weakly Ramsey but not Ramsey.

All ultrafilters Tukey reducible to $_1$ are countable Fubini products of a fixed collection.

There is a unique nonprincipal ultrafilter Tukey type below $_1$, the Ramsey ultrafilter.

Abstract

Motivated by a Tukey classification problem we develop here a new topological Ramsey space $\mathcal{R}_1$ that in its complexity comes immediately after the classical is a natural Ellentuck space \cite{MR0349393}. Associated with $\mathcal{R}_1$ is an ultrafilter $\mathcal{U}_1$ which is weakly Ramsey but not Ramsey. We prove a canonization theorem for equivalence relations on fronts on $\mathcal{R}_1$. This is analogous to the Pudlak-\Rodl\ 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 $\mathcal{U}_1$: Every ultrafilter which is Tukey reducible to $\mathcal{U}_1$ 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 $\mathcal{U}_1$, namely the Tukey type a Ramsey ultrafilter.