A Ramsey-Classification Theorem and its Application in the Tukey Theory of Ultrafilters
Natasha Dobrinen, Stevo Todorcevic

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 that in its complexity comes immediately after the classical is a natural Ellentuck space \cite{MR0349393}. Associated with is an ultrafilter which is weakly Ramsey but not Ramsey. We prove a canonization theorem for equivalence relations on fronts on . 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 : Every ultrafilter which is Tukey reducible to is isomorphic to a countable iteration of Fubini products of ultrafilters from among a fixed countable collection of ultrafilters. Moreover, we show…
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Taxonomy
TopicsAdvanced Topology and Set Theory · Limits and Structures in Graph Theory · Computability, Logic, AI Algorithms
