On the subset Combinatorics of G-spaces

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 $\mathcal{J}$ 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 $βX$ of the discrete space $X$, we characterize the points $p\inβX$ 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.

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