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