Paper detail

A notion of selective ultrafilter corresponding to topological Ramsey spaces

We introduce the relation of "almost-reduction" in an arbitrary topological Ramsey space R, as a generalization of the relation of "almost-inclusion" on the space of infinite sets of natural numbers (the Ellentuck space). This leads us to a type of ultrafilter U on the set of first approximations of the elements of R which corresponds to the well-known notion of "selective ultrafilter" on N, the set of natural numbers. The relationship turns out to be rather exact in the sense that it permits us to lift several well-known facts about selective ultrafilters on N and the Ellentuck space to the ultrafilter U and the Ramsey space R. For example, we prove that the Open Coloring Axiom holds in M[U], where M is a Solovay model. In this way we extend a result due to Di Prisco and Todorcevic which gives the same conclusion for the Ellentuck space.