This article is cited in 6 papers

RESEARCH ARTICLE

Partitions of groups into thin subsets

Igor Protasov

Department of Cybernetics, Kyiv National University, Volodymyrska 64, 01033, Kyiv, Ukraine

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.

Keywords: $\kappa$-thin subsets of a group, partition of a group.

MSC: 03E75, 20F99, 20K99

Received: 13.03.2011
Revised: 13.03.2011

Language: English

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