This article is cited in 2 papers

RESEARCH ARTICLE

Partitions of groups into sparse subsets

Igor Protasov

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

Abstract: A subset $A$ of a group $G$ is called sparse if, for every infinite subset $X$ of $G$, there exists a finite subset $F\subset X$, such that $\bigcap_{x\in F} xA$ is finite. We denote by $\eta(G)$ the minimal cardinal such that $G$ can be partitioned in $\eta(G)$ sparse subsets. If $|G| > (\kappa^+)^{\aleph_0}$ then $\eta(G) > \kappa$, if $|G|\leqslant \kappa^+$ then $\eta(G) \leqslant \kappa$. We show also that $cov(A) \geqslant cf|G|$ for each sparse subset $A$ of an infinite group $G$, where $cov(A)=\min\{|X|: G = XA\}$.

Keywords: partition of a group, sparse subset of a group.

MSC: 03E75, 20F99, 20K99

Language: English

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