Local Ramsey theory. An abstract approach

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.