Local Ramsey theory. An abstract approach

TL;DR

This paper develops an abstract framework for topological Ramsey spaces, extending the notion of semiselective coideals, and explores their implications for forcing and definability, generalizing classical results in Ramsey theory.

Contribution

It introduces a generalized approach to topological Ramsey spaces, extending semiselective coideal concepts and analyzing their forcing and definability properties in a broad context.

Findings

Semiselective ultrafilters are generic over L(ℝ) under large cardinal hypotheses.

Every definable subset of a topological Ramsey space is H-Ramsey for semiselective coideals.

Generalization of classical Ramsey results to a broad class of topological Ramsey spaces.

Abstract

Given a topological Ramsey space $(\mathcal R,\leq, r)$, we extend the notion of semiselective coideal to sets $\mathcal H\subseteq\mathcal R$ and study conditions for $\mathcal H$ that will enable us to make the structure $(\mathcal R,\mathcal H,\leq, r)$ a Ramsey space (not necessarily topological) and also study forcing notions related to $\mathcal H$ which will satisfy abstract versions of interesting properties of the corresponding forcing notions in the realm of Ellentuck's space. This extends results of Farah, and results of Mijares, to the most general context of topological Ramsey spaces. As applications, we prove that for every topological Ramsey space $\mathcal R$, under suitable large cardinal hypotheses every semiselective ultrafilter $\mathcal U\subseteq\mathcal R$ is generic over $L(\mathbb R)$; and that given a semiselective coideal $\mathcal H\subseteq\mathcal R$, every definable subset of $\mathcal R$ is $\mathcal H$--Ramsey. This generalizes the corresponding results for the case when $\mathcal R$ is equal to Ellentuck's space.