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