Geodesic
On the subset combinatorics of $G$-spaces
Algebra and discrete mathematics, Tome 17 (2014) no. 1, pp. 98-109.

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Let $G$ be a group and let $X$ be a transitive $G$-space. We classify the subsets of $X$ with respect to a translation invariant ideal $J$ in the Boolean algebra of all subsets of $X$, introduce and apply the relative combinatorical derivations of subsets of $X$. Using the standard action of $G$ on the Stone-Čech compactification $\beta X$ of the discrete space $X$, we characterize the points $p\in\beta X$ isolated in $Gp$ and describe a size of a subset of $X$ in terms of its ultracompanions in $\beta X$. We introduce and characterize scattered and sparse subsets of $X$ from different points of view.
Keywords: $G$-space, relative combinatorial derivation, Stone-Čech compactification, ultracompanion, sparse and scattered subsets. 
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Igor Protasov; Sergii Slobodianiuk. On the subset combinatorics of $G$-spaces. Algebra and discrete mathematics, Tome 17 (2014) no. 1, pp. 98-109. http://geodesic.mathdoc.fr/item/ADM_2014_17_1_a6/

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