On the subset combinatorics of \(G\)-spaces

Abstract

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-\(\check{C}\)ech 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.