Slobodianiuk S. Combinatorics of subsets of groups

It is shown that each group G admits partition of cardunality |G| on w_1-large subsets. It is proved that for any partition of a group group on n subsets, one of the parts must be n!-dense. It is proved that each group for each positive integer k admits a partition into two k-meager sets. It is proved that an arbitrary m-thin set of arbitrary Abelian group of cardinality w_1 admits a partition into m thin subsets, while for larger cardinality this is not true. It is proved that there exists a kaleidoscopical configuration in any infinite Abelian group and in the Euclidean space.