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 (cf. [9][E. Ellentuck, 1974], [19][A. R. D. Mathias, 1977]). This extends results from [10][I. Farah, 1998], [20][J. G. Mijares, 2007] 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 coideal H ⊆ R , every definable subset of R is H ‐Ramsey. This generalizes the corresponding results for the case when R is equal to Ellentuck's space (cf. [3][C. Di Prisco, 2012], [10][I. Farah, 1998]).