Ramsey for $\mathcal{R}_{1}$ ultrafilter mappings and their Dedekind cuts

Abstract

Associated to each ultrafilter $\mathcal{U}$ on $\omega$ and each map $p:\omega\rightarrow \omega$ is a Dedekind cut in the ultrapower $\omega^{\omega}/p( \mathcal{U})$. Blass has characterized, under CH, the cuts obtainable when $\mathcal{U}$ is taken to be either a p-point ultrafilter, a weakly-Ramsey ultrafilter or a Ramsey ultrafilter. Dobrinen and Todorcevic have introduced the topological Ramsey space $\mathcal{R}_{1}$. Associated to the space $\mathcal{R}_{1}$ is a notion of Ramsey ultrafilter for $\mathcal{R}_{1}$ generalizing the familiar notion of Ramsey ultrafilter on $\omega$. We characterize, under CH, the cuts obtainable when $\mathcal{U}$ is taken to be a Ramsey for $\mathcal{R}_{1}$ ultrafilter and $p$ is taken to be any map. In particular, we show that the only cut obtainable is the standard cut, whose lower half consists of the collection of equivalence classes of constants maps. Forcing with $\mathcal{R}_{1}$ using almost-reduction adjoins an ultrafilter which is Ramsey for $\mathcal{R}_{1}$. For such ultrafilters $\mathcal{U}_{1}$, Dobrinen and Todorcevic have shown that the Rudin-Keisler types of the p-points within the Tukey type of $\mathcal{U}_{1}$ consists of a strictly increasing chain of rapid p-points of order type $\omega$. We show that for any Rudin-Keisler mapping between any two p-points within the Tukey type of $\mathcal{U}_{1}$ the only cut obtainable is the standard cut. These results imply existence theorems for special kinds of ultrafilters.