93 research outputs found
On the subset Combinatorics of G-spaces
Let be a group and let be a transitive -space. We classify the
subsets of with respect to a translation invariant ideal in
the Boolean algebra of all subsets of , introduce and apply the relative
combinatorical derivations of subsets of . Using the standard action of
on the Stone-ech compactification of the discrete space
, we characterize the points isolated in and describe a
size of a subset of in terms of its ultracompanions in . We
introduce and characterize scattered and sparse subsets of from different
points of view
Ultrafilters on -spaces
For a discrete group and a discrete -space , we identify the
Stone-\v{C}ech compactifications and with the sets of all
ultrafilters on and , and apply the natural action of on
to characterize large, thick, thin, sparse and scattered subsets of
. We use -invariant partitions and colorings to define -selective and
-Ramsey ultrafilters on . We show that, in contrast to the
set-theoretical case, these two classes of ultrafilters are distinct. We
consider also universally thin ultrafilters on , the -points, and
study interrelations between these ultrafilters and some classical ultrafilters
on
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