  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 $\beta X$ 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.
