Selective and Ramsey ultrafilters on $G$-spaces

O. V. Petrenko, I. V. Protasov

Let $G$ be a group, $X$ be an infinite transitive $G$-space. A free ultrafilter $\UU$ on $X$ is called $G$-selective if, for any $G$-invariant partition $\PP$ of $X$, either one ce…