Selective and Ramsey Ultrafilters on G-spaces

Abstract

Let $G$ be a group, and let $X$ be an infinite transitive $G$-space. A free ultrafilter $\mathcal{U}$ on $X$ is called $G$-selective if, for any $G$-invariant partition $\mathcal{P}$ of $X$, either one cell of $\mathcal{P}$ is a member of $\mathcal{U}$, or there is a member of $\mathcal{U}$ which meets each cell of $\mathcal{P}$ in at most one point. We show that in ZFC with no additional set-theoretical assumptions there exists a $G$-selective ultrafilter on $X$. We describe all $G$-spaces $X$ such that each free ultrafilter on $X$ is $G$-selective, and we prove that a free ultrafilter $\mathcal{U}$ on $\omega$ is selective if and only if $\mathcal{U}$ is $G$-selective with respect to the action of any countable group $G$ of permutations of $\omega$.

A free ultrafilter $\mathcal{U}$ on $X$ is called $G$-Ramsey if, for any $G$-invariant coloring $\chi :[X{]}^2\to \{0,1\}$, there is $U\in \mathcal{U}$ such that $[U{]}^2$ is $\chi$-monochromatic. We show that each $G$-Ramsey ultrafilter on $X$ is $G$-selective. Additional theorems give a lot of examples of ultrafilters on $\mathbb{Z}$ that are $\mathbb{Z}$-selective but not $\mathbb{Z}$-Ramsey.