Ultrafilters on $G$-spaces

For a discrete group $G$ and a discrete $G$-space $X$, we identify the Stone-Čech compactifications $\beta G$ and $\beta X$ with the sets of all ultrafilters on $G$ and $X$, and apply the natural action of $\beta G$ on $\beta X$ to characterize large, thick, thin, sparse and scattered subsets of $X$. We use $G$-invariant partitions and colorings to define $G$-selective and $G$-Ramsey ultrafilters on $X$. We show that, in contrast to the set-theoretical case, these two classes of ultrafilters are distinct. We consider also universally thin ultrafilters on $\omega$, the $T$-points, and study interrelations between these ultrafilters and some classical ultrafilters on $\omega$.