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Constructing subsets of a given packing index in Abelian groups. (English) Zbl 1160.20315

Summary: By definition, the sharp packing index \(\text{ind}_P^\#(A)\) of a subset \(A\) of an Abelian group \(G\) is the smallest cardinal \(\kappa\) such that for any subset \(B\subset G\) of size \(|B|\geq\kappa\) the family \(\{b+A:b\in B\}\) is not disjoint. We prove that an infinite Abelian group \(G\) contains a subset \(A\) with given index \(\text{ind}_P^\#(A)=\kappa\) if and only if one of the following conditions holds: (1) \(2\leq\kappa\leq|G|^+\) and \(k\notin\{3,4\}\); (2) \(\kappa=3\) and \(G\) is not isomorphic to \(\bigoplus_{i\in I}\mathbb{Z}_3\); (3) \(\kappa=4\) and \(G\) is not isomorphic to \(\bigoplus_{i\in I}\mathbb{Z}_2\) or to \(\mathbb{Z}_4\oplus(\bigoplus_{i\in I}\mathbb{Z}_2)\).

MSC:

20K99 Abelian groups
52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
05D05 Extremal set theory
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