Asymmetric decompositions of Abelian groups (Q1972538)

A subset \(A\) of an Abelian group \(G\) is said to be asymmetric if \(g+S\not\subset A\) for any element \(g\in G\) and any infinite symmetric subset \(S\subset G\) (\(S=-S\)). The minimal cardinality of a decomposition of the group \(G\) into asymmetric sets is denoted by \(\nu(G)\). For any Abelian group \(G\), the cardinal number \(\nu(G)\) is expressed via the following cardinal invariants: the free rank, the 2-rank, and the cardinality of the group. In particular, \(\nu(\mathbb{Z}^n)=n+1\), \(\nu(\mathbb{Q}^n)=n+2\), and \(\nu(\mathbb{R})=\aleph_0\).