Monochrome Symmetric Subsets in 2-Colorings of Groups

Abstract

A subset $A$ of a group $G$ is called symmetric with respect to the element $g\in G$ if $A=gA^{-1}g$. It is proved that in any 2-coloring, every infinite group $G$ contains monochrome symmetric subsets of arbitrarily large cardinality $ < |G|$.