Geodesic
Partitions of groups into thin subsets
Algebra and discrete mathematics, Tome 11 (2011) no. 2, pp. 78-81

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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
Mots-clés : partition of a group. 
@article{ADM_2011_11_2_a4,
     author = {Igor Protasov},
     title = {Partitions of groups into thin subsets},
     journal = {Algebra and discrete mathematics},
     pages = {78--81},
     publisher = {mathdoc},
     volume = {11},
     number = {2},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2011_11_2_a4/}
}
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Igor Protasov. Partitions of groups into thin subsets. Algebra and discrete mathematics, Tome 11 (2011) no. 2, pp. 78-81. http://geodesic.mathdoc.fr/item/ADM_2011_11_2_a4/