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The dynamical look at the subsets of a group. (English) Zbl 1329.54038

Some analogues of notions from topological dynamics are formulated in the setting of infinite combinatorics on groups, and various relationships between them examined. Given a group \(G\), write \(P(G)\), \([G]^{<\omega}\), \([G]^n\) for the set of all subsets, finite subsets, subsets of cardinality \(n\) of \(G\) respectively. A subset \(A\subset G\) is called thin if \(A\cap Ag\in[G]^{<\omega}\) for all \(g\in G\setminus\{e\}\); \(n\)-thin if \(g_0A\cap\cdots\cap g_nA\in[G]^{<\omega}\) for any distinct \(g_0,\dots,g_n\in G\); sparse if for every infinite subset \(X\) of \(G\), there is a finite subset \(F\) of \(X\) for which \(\cap_{g\in F}gA\in[G]^{<\omega}\); scattered if for any \(B\subset A\) there is some \(F\in[G]^{<\omega}\) such that for each \(H\in[G]^{<\omega}\) with \(F\cap H=\emptyset\) there is some \(b\in B\) such that \(Hb\cap B=\emptyset\); and thick if for any \(F\in[G]^{<\omega}\) there exists some \(g\in G\) with \(Fg\subset A\). Here dynamical characterizations (expressed by identifying \(P(G)\) with \(\{0,1\}^G\) in the compact product topology) are given of all these notions. It is known that any \(n\)-thin subset of a group with 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. Here the \(\aleph_1\) case is clarified, showing that each \(n\)-thin subset of an abelian group of cardinality \(\aleph_1\) can be partitioned into \(n\) \(1\)-thin subsets.

MSC:

54H20 Topological dynamics (MSC2010)
05C15 Coloring of graphs and hypergraphs

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