Partitions of groups into thin subsets. (English) Zbl 1257.20046

Protasov, Igor

Algebra Discrete Math. 11, No. 2, 78-81 (2011).

Summary: Let \(G\) be an infinite group with the identity \(e\), \(\kappa\) be an infinite cardinal \(\leq|G|\). A subset \(A\subset G\) is called \(\kappa\)-thin if \(|gA\cap A|\leq\kappa\) 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.

MSC:

20F99	Special aspects of infinite or finite groups
03E75	Applications of set theory
03E50	Continuum hypothesis and Martin’s axiom
20F05	Generators, relations, and presentations of groups
05D10	Ramsey theory

Keywords:

infinite groups; thin subsets of groups; partitions of groups