Selective but not Ramsey

Abstract

We give a partial answer to the following question of Dobrinen: For a given topological Ramsey space $\mathcal{R}$, are the notions of selective for $\mathcal{R}$ and Ramsey for $\mathcal{R}$ equivalent? Every topological Ramsey space $\mathcal{R}$ has an associated notion of Ramsey ultrafilter for $\mathcal{R}$ and selective ultrafilter for $\mathcal{R}$ (see \cite{MijaresSelective}). If $\mathcal{R}$ 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 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, and in fact a countable collection of such examples. For each positive integer $n$ we show that for the topological Ramsey space $\mathcal{R}_{n}$ from \cite{Ramsey-Class2}, the notions of selective for $\mathcal{R}_{n}$ and Ramsey for $\mathcal{R}_{n}$ 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 $\mathcal{R}_{n}$. Moreover, we introduce a notion of finite product among members of the family $\{\mathcal{R}_{n}: n<ω\}$. 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.