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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 cutsobtainable when $\mathcal{U}$ is taken to be either a p-point ultrafilter, aweakly-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 ofRamsey ultrafilter for $\mathcal{R}_{1}$ generalizing the familiar notion ofRamsey ultrafilter on $\omega$. We characterize, under CH, the cuts obtainablewhen $\mathcal{U}$ is taken to be a Ramsey for $\mathcal{R}_{1}$ ultrafilterand $p$ is taken to be any map. In particular, we show that the only cutobtainable is the standard cut, whose lower half consists of the collection ofequivalence classes of constants maps.  Forcing with $\mathcal{R}_{1}$ using almost-reduction adjoins an ultrafilterwhich 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-pointswithin the Tukey type of $\mathcal{U}_{1}$ consists of a strictly increasingchain of rapid p-points of order type $\omega$. We show that for anyRudin-Keisler mapping between any two p-points within the Tukey type of$\mathcal{U}_{1}$ the only cut obtainable is the standard cut. These resultsimply existence theorems for special kinds of ultrafilters.

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