Selective but not Ramsey

We give a partial answer to the following question of Dobrinen: For a given topological Ramsey space RR, are the notions of selective for RR and Ramsey for RR equivalent? Every topological Ramsey space RR has an associated notion of Ramsey ultrafilter for RR and selective ultrafilter for RR (see [1]). If RR is taken to be the Ellentuck space then the two concepts reduce to the familiar notions of Ramsey and selective ultrafilters on ω; so by a well-known result of Kunen the two are equivalent. We give the first example of an ultrafilter on a topological Ramsey space that is selective but not Ramsey for the space.We show that for the topological Ramsey space R1R1 from [2], the notions of selective for R1R1 and Ramsey for R1R1 are not equivalent. In particular, we prove that forcing with a closely related topological Ramsey space using almost-reduction, adjoins an ultrafilter that is selective but not Ramsey for R1R1.