Ultrafilters on $G$-spaces

For a discrete group $G$ and a discrete $G$-space $X$, we identify the Stone-Čech compactifications $βG$ and $βX$ with the sets of all ultrafilters on $G$ and $X$, and apply the natural action of $βG$ on $βX$ 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 $ω$, the $T$-points, and study interrelations between these ultrafilters and some classical ultrafilters on $ω$.

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