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. 
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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/

[1] T. Banakh, O. Chervak, L. Zdomskyy, “On character of points in the Higson corona of a metric space”, Comment. Math. Univ. Carolin., 54:2 (2013), 159–178 | MR | Zbl

[2] T. Banakh, I. Protasov, S. Slobodianiuk, “Densities, submeasures and partitions of groups”, Algebra Discrete Math., 17:2 (2014), 193–221 | MR | Zbl

[3] T. Banakh, I. Protasov, S. Slobodianiuk, “Scattered subsets of groups”, Ukr. Math. J., 67 (2015), 304–312 | DOI | MR

[4] W. W. Comfort, “Ultrafilters: some old and some new results”, Bull. Amer. Math. Soc., 83 (1977), 417–455 | DOI | MR | Zbl

[5] H. Dales, A. Lau, D. Strauss, “Banach Algebras on a Semigroups and Their Compactifications”, Mem. Amer. Math. Soc., 2005, 2010 | MR

[6] M. Filali, I. Protasov, Ultrafilters and Topologies on Groups, Math. Stud. Monogr. Ser., 13, VNTL Publishers, Lviv, 2011

[7] N. Hindman, D. Strauss, Algebra in the Stone-Čech compactification: Theory and Applications, Walter de Gruyter, Berlin–New York, 1998 | MR | Zbl

[8] O. Petrenko, I. Protasov, “Thin ultrafilters”, Notre Dame J. Formal Logic, 53 (2012), 79–88 | DOI | MR | Zbl

[9] O. Petrenko, I. Protasov, “Selective and Ramsey ultrafilters on $G$-spaces”, Notre Dame J. Formal Logic (to appear)

[10] O. V. Petrenko, I. V. Protasov, “Balleans and $G$-spaces”, Ukr. Math. J., 64 (2012), 344–350 | DOI | MR | Zbl

[11] O. V. Petrenko, I. V. Protasov, “Balleans and filters”, Matem. Stud., 38 (2012), 3–11 | MR | Zbl

[12] I. Protasov, Combinatorics of Numbers, Math. Stud. Monogr. Ser, 2, VNTL Publisher, Lviv, 1997 | MR | Zbl

[13] I. Protasov, “Coronas of balleans”, Topology Appl., 149 (2005), 149–160 | DOI | MR | Zbl

[14] I. Protasov, “Dynamical equivalences on $G^*$”, Topology Appl., 155 (2008), 1394–1402 | DOI | MR | Zbl

[15] I. V. Protasov, “Balleans of bounded geometry and $G$-spaces”, Mat. Stud., 30 (2008), 61–66 | MR | Zbl

[16] I. Protasov, “Counting $\Omega$-ideals”, Algebra Universalis, 62 (2010), 339–343 | DOI | MR

[17] I. Protasov, “Coronas of ultrametric spaces”, Comment. Math. Univ. Carolin., 52:2 (2011), 303–307 | MR | Zbl

[18] I. Protasov, “Ultrafilters and partitions of Abelian groups”, Ukr. Math. J., 53 (2011), 99–107 | DOI | MR

[19] I. Protasov, “Weak $P$-points in coronas of $G$-spaces”, Topology Appl., 159 (2012), 587–592 | DOI | MR | Zbl

[20] I. Protasov, “Ultrafilters on metric spaces”, Topology Appl., 164 (2014), 207–214 | DOI | MR | Zbl

[21] I. Protasov, T. Banakh, Ball Structures and Colorings of Graphs and Groups, Math. Stud. Monogr. Ser, 11, VNTL Publisher, Lviv, 2003 | MR | Zbl

[22] I. Protasov, S. Slobodianiuk, “Partitions of groups”, Math. Stud., 42 (2014), 115–128 | MR | Zbl

[23] I. Protasov, S. Slobodianiuk, “On the subset combinatorics of $G$-spaces”, Algebra Discrete Math., 17:1 (2014), 98–109 | MR | Zbl

[24] I. Protasov, S. Slobodianiuk, “Ultrafilters on balleans”, Ukr. Math. J. (to appear)

[25] I. Protasov, M. Zarichnyi, General Asymptology, Math. Stud. Monogr. Ser., 12, VNTL Publishers, Lviv, 2007 | MR | Zbl

[26] J. Roe, Lectures on Coarse Geometry, Amer. Math. Soc., Providence, R. I., 2003 | MR | Zbl

[27] S. Todorcevic, Introduction to Ramsey Spaces, Princeton Univ. Press, 2010 | MR

[28] Y. Zelenyuk, Ultrafilters and Topologies on Groups, de Grueter, 2012 | MR