15.77 (2002)

Elements $a, b$ of a group $G$ are said to be symmetric with respect to an element $g \in G$ if $a = g b^{-1} g$. Let $G$ be an infinite abelian group, $\alpha$ a cardinal, $\alpha < |G|$. Is it true that for any $n$-colouring $\chi : G \to \{0, 1, \dots, n-1\}$ there exists a monochrome subset of cardinality $\alpha$ that is symmetric with respect to some element of $G$? This is known to be true for $n \leqslant 3$.