Infinite dimensional Ellentuck spaces and Ramsey-classification theorems

TL;DR

This paper extends finite-dimensional Ellentuck spaces to infinite dimensions, constructing a hierarchy of topological Ramsey spaces based on uniform barriers, and proves new Ramsey-classification theorems that generalize existing results.

Contribution

It introduces a continuum of infinite-dimensional Ellentuck-like spaces based on uniform barriers and establishes new Ramsey-classification theorems extending the Pudlak-Rodl Theorem.

Findings

Constructed continuum many topological Ramsey spaces $\\mathcal{E}_B$

Proved Ramsey-classification theorems for equivalence relations on these spaces

Extended the Pudlak-Rodl Theorem to barriers on the new spaces

Abstract

We extend the hierarchy of finite-dimensional Ellentuck spaces to infinite dimensions. Using uniform barriers $B$ on $\omega$ as the prototype structures, we construct a class of continuum many topological Ramsey spaces $\mathcal{E}_B$ which are Ellentuck-like in nature, and form a linearly ordered hierarchy under projection. We prove new Ramsey-classification theorems for equivalence relations on fronts, and hence also on barriers, on the spaces $\mathcal{E}_B$, extending the Pudlak-Rodl Theorem for barriers on the Ellentuck space. The inspiration for these spaces comes from continuing the iterative construction of the forcings $\mathcal{P}([\om]^k)/\Fin^{\otimes k}$ to the countable transfinite. The $\sigma$-closed partial order $(\mathcal{E}_B, \sse^{\Fin^{B}})$ is forcing equivalent to $\mathcal{P}(B)/\Fin^{B}$, which forces a non-p-point ultrafilter $\mathcal{G}_B$. The present work forms the basis for further work classifying the Rudin-Keisler and Tukey structures for the hierarchy of the generic ultrafilters $\mathcal{G}_B$.