Local Ramsey theory: an abstract approach

Given a topological Ramsey space (R,≤, r), we extend the notion of semiselective coideal to sets H ⊆ R and study conditions for H that will enable us to make the structure (R,H,≤, r) a Ramsey space (not necessarily topological) and also study forcing notions related to H which will satisfy abstract versions of interesting properties of the corresponding forcing notions in the realm of Ellentuck’s space (see [7, 15]). This extends results from [8, 17] to the most general context of topological Ramsey spaces. As applications, we prove that for every topological Ramsey space R, under suitable large cardinal hypotheses every semiselective ultrafilter U ⊆ R is generic over L(R); and that given a semiselective coidealH ⊆ R, every definable subset ofR isH–Ramsey. This generalizes the corresponding results for the case when R is equal to Ellentuck’s space (see [8, 3]).