Selective but not Ramsey

We give a partial answer to the following question of Dobrinen: For a given topological Ramsey space R, are the notions of selective for R and Ramsey for R equivalent? Every topological Ramsey space R has an associated notion of Ramsey ultrafilter for R and selective ultrafilter for R (see [1]). If R 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 wellknown 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, and in fact a countable collection of such examples. For each positive integer n we show that for the topological Ramsey space Rn from [2], the notions of selective for Rn and Ramsey for Rn are not equivalent. In particular, we prove that forcing with a closely related space using almost-reduction, adjoins an ultrafilter that is selective but not Ramsey for Rn. Moreover, we introduce a notion of finite product among members of the family {Rn : n < ω}. We show that forcing with closely related product spaces using almost-reduction, adjoins ultrafilters that are selective but not Ramsey for these product topological Ramsey spaces.