Ramsey for R1 ultrafilter mappings and their Dedekind cuts

Associated to each ultrafilter U on ω and each map p : ω → ω is a Dedekind cut in the ultrapower ωω/p(U). Blass has characterized, under CH, the cuts obtainable when 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 R1. Associated to the space R1 is a notion of Ramsey ultrafilter for R1 generalizing the familiar notion of Ramsey ultrafilter on ω. We characterize, under CH, the cuts obtainable when U is taken to be a Ramsey for R1 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 R1 using almost-reduction adjoins an ultrafilter which is Ramsey for R1. For such ultrafilters U1, Dobrinen and Todorcevic have shown that the Rudin-Keisler types of the p-points within the Tukey type of U1 consists of a strictly increasing chain of rapid p-points of order type ω. We show that for any Rudin-Keisler mapping between any two p-points within the Tukey type of U1 the only cut obtainable is the standard cut. These results imply existence theorems for special kinds of ultrafilters.