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-ˇCech compactification βX of the discrete space X, we characterize the points p ∈ βX isolated in Gp and describe a size of a subset of X in terms of its ultracompanions in βX. We introduce and characterize scattered and sparse subsets of X from different points of view.