Ultrafilters on $G$-spaces

TL;DR

This paper explores ultrafilters on $G$-spaces, characterizing various large and sparse subsets via the Stone-ch compactification, and introduces new classes of ultrafilters with distinct properties.

Contribution

It identifies ultrafilters on $G$-spaces with Stone-ch compactifications and introduces $G$-selective and $G$-Ramsey ultrafilters, highlighting their differences from classical ultrafilters.

Findings

Characterization of large, thick, thin, sparse, scattered subsets using ultrafilters.

Introduction of $G$-selective and $G$-Ramsey ultrafilters and their distinction from classical ultrafilters.

Analysis of universally thin ultrafilters and their relation to classical ultrafilters on a9.

Abstract

For a discrete group $G$ and a discrete $G$-space $X$, we identify the Stone-\v{C}ech 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 $\omega$.