  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 cell of $\PP$ is a member of $\UU$, or there
is a member of $\UU$ which meets each cell of $\PP$ in at most one point. We
show (Theorem 1) that in ZFC with no additional set-theoretical assumptions
there exists a $G$-selective ultrafilter on $X$, describe all $G$-spaces $X$
(Theorem 2) such that each free ultrafilter on $X$ is $G$-selective, and prove
(Theorem 3) that a free ultrafilter $\UU$ on $\omega$ is selective if and only
if $\UU$ is $G$-selective with respect to the action of any countable group $G$
of permutations of $\omega$.
  A free ultrafilter $\UU$ on $X$ is called $G$-Ramsey if, for any
$G$-invariant coloring $\chi:[G]^2 \to \{0,1\}$, there is $U\in \UU$ such that
$[U]^2$ is $\chi$-monochrome. By Theorem 4, each $G$-Ramsey ultrafilter on $X$
is $G$-selective. Theorems 5 and 6 give us a plenty of $\mathbb{Z}$-selective
ultrafilters on $\mathbb{Z}$ (as a regular $\mathbb{Z}$-space) but not
$\mathbb{Z}$-Ramsey. We conjecture that each $\mathbb{Z}$-Ramsey ultrafilter is
selective.
