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 Algebra Discrete Math.: Year: Volume: Issue: Page: Find

 Algebra Discrete Math., 2012, Volume 13, Issue 1, Pages 107–110 (Mi adm68)

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.

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MSC: 03E75, 20F99, 20K99
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Citation: Igor Protasov, “Partitions of groups into sparse subsets”, Algebra Discrete Math., 13:1 (2012), 107–110

Citation in format AMSBIB
\Bibitem{Pro12} \by Igor~Protasov \paper Partitions of groups into sparse subsets \jour Algebra Discrete Math. \yr 2012 \vol 13 \issue 1 \pages 107--110 \mathnet{http://mi.mathnet.ru/adm68} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2963828} \zmath{https://zbmath.org/?q=an:1258.20036} 

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This publication is cited in the following articles:
1. Banakh T.O. Protasov I.V. Slobodianiuk S.V., “Scattered Subsets of Groups”, Ukr. Math. J., 67:3 (2015), 347–356
2. Protasov I. Slobodianiuk S., “Partitions of Groups”, Adv. Appl. Discret. Math., 15:1 (2015), 33–60
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