Geodesic
Prethick subsets in partitions of groups
Algebra and discrete mathematics, Tome 14 (2012) no. 2, pp. 267-275

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A subset $S$ of a group $G$ is called thick if, for any finite subset $F$ of $G$, there exists $g\in G$ such that $Fg\subseteq S$, and $k$-prethick, $k\in \mathbb{N}$ if there exists a subset $K$ of $G$ such that $|K|=k$ and $KS$ is thick. For every finite partition $\mathcal{P}$ of $G$, at least one cell of $\mathcal{P}$ is $k$-prethick for some $k\in \mathbb{N}$. We show that if an infinite group $G$ is either Abelian, or countable locally finite, or countable residually finite then, for each $k\in \mathbb{N}$, $G$ can be partitioned in two not $k$-prethick subsets.
Keywords: thick and $k$-prethick subsets of groups, $k$-meager partition of a group. 
@article{ADM_2012_14_2_a9,
     author = {Igor Protasov and Sergiy Slobodianiuk},
     title = {Prethick subsets in partitions of groups},
     journal = {Algebra and discrete mathematics},
     pages = {267--275},
     publisher = {mathdoc},
     volume = {14},
     number = {2},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2012_14_2_a9/}
}
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Igor Protasov; Sergiy Slobodianiuk. Prethick subsets in partitions of groups. Algebra and discrete mathematics, Tome 14 (2012) no. 2, pp. 267-275. http://geodesic.mathdoc.fr/item/ADM_2012_14_2_a9/