Partitions of groups into thin subsets

Abstract

Let \(G\) be an infinite group with the identity \(e\), \(\kappa\) be an infinite cardinal \(\leqslant |G|\). A subset \(A\subset G\) is called \(\kappa\)-thin if \(|gA\cap A|\leqslant\kappa\) for every \(g\in G\setminus\{e\}\). We calculate the minimal cardinal \(\mu(G,\kappa)\) such that \(G\) can be partitioned in \(\mu(G,\kappa)\) \(\kappa\)-thin subsets. In particular, we show that the statement \(\mu(\mathbb{R},\aleph_0)=\aleph_0\) is equivalent to the Continuum Hypothesis.