The dynamical look at the subsets of a group

Igor V. Protasov, Sergii Slobodianiuk

Abstract

We consider the action of a group $G$ on the family $\mathcal{P}(G)$ of all subsets of $G$ by the right shifts $A\mapsto Ag$ and give the dynamical characterizations of thin, $n$-thin, sparse and scattered subsets.

For $n\in\mathbb{N}$, a subset $A$ of a group $G$ is called $n$-thin if $g_0A\cap\dots\cap g_nA$ is finite for all distinct $g_0,\dots,g_n\in G$.
Each $n$-thin subset of a group of cardinality $\aleph_0$ can be partitioned into $n$ $1$-thin subsets but there is a $2$-thin subset in some Abelian group of cardinality $\aleph_2$ which cannot be partitioned into two $1$-thin subsets. We eliminate the gap between $\aleph_0$ and $\aleph_2$ proving that each $n$-thin subset of an Abelian group of cardinality $\aleph_1$ can be partitioned into $n$ $1$-thin subsets.


Keywords

Thin; sparse and scatterad subsets of a group; recurrent point; chromatic number of a graph.

Subject classification

54H20; 05C15.

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References

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1. Recent progress in subset combinatorics of groups
Igor V. Protasov, Ksenia D. Protasova
Journal of Mathematical Sciences  vol: 234  issue: 1  first page: 49  year: 2018  
doi: 10.1007/s10958-018-3980-0



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Universitat Politècnica de València

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