Prethick subsets in partitions of groups

Abstract

A subset S of a group G is called thick if, for any finite subset F of G, there exists g ∈ G such that Fg ⊆ S, and k-prethick, k ∈ N if there exists a subset K of G such that |K| = k and KS is thick. For every finite partition P of G, at least one cell of P is k-prethick for some k ∈ N. We show that if an infinite group G is either Abelian, or countable locally finite, or countable residually finite then, for each k ∈ N, G can be partitioned in two not k-prethick subsets.