On a cardinal group invariant related to decompositions of Abelian groups

For each Abelian group $G$, a cardinal invariant $\chi(G)$ is introduced and its properties are studied. In the special case $G=\mathbb Z^n$, the cardinal $\chi\mathbb Z^n)$ is equal to the minimal cardinality of an essential subset of $\mathbb Z^n$, i.e., a of a subset $A\subset\mathbb Z^n$ such that, for any coloring of the group $\mathbb Z^n$ in $n$ colors, there exists an infinite one-color subset that is symmetric with respect to some point $\alpha$ of $A$. The estimate $n(n+1)/2\le\chi(\mathbb Z^n)<2^n$ is proved for all $n$ and the relation $\chi(\mathbb Z^n)=n(n+1)/2$ for $n\le3$. The structure of essential subsets of cardinality $\chi(\mathbb Z^n)$ in $\mathbb Z^n$ is completely described for $n\le3$.