Geodesic
Ultrafilters on $G$-spaces
Algebra and discrete mathematics, Tome 19 (2015) no. 2, pp. 254-269

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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 $\omega$.
Keywords: $G$-space, ultrafilters, ultracompanion, $G$-selective ultrafilter, $G$-Ramsey ultrafilter, $T$-point, ballean, asymorphism. 
@article{ADM_2015_19_2_a8,
     author = {O. V. Petrenko and I. V. Protasov},
     title = {Ultrafilters on $G$-spaces},
     journal = {Algebra and discrete mathematics},
     pages = {254--269},
     publisher = {mathdoc},
     volume = {19},
     number = {2},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2015_19_2_a8/}
}
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O. V. Petrenko; I. V. Protasov. Ultrafilters on $G$-spaces. Algebra and discrete mathematics, Tome 19 (2015) no. 2, pp. 254-269. http://geodesic.mathdoc.fr/item/ADM_2015_19_2_a8/