17.102 (2010)

We say that two subsets $A, B$ of an infinite group $G$ are separated if there exists an infinite subset $X$ of $G$ such that $1 \in X$, $X = X^{-1}$, and $XAX \cap B = \varnothing$. Is it true that any two disjoint subsets $A, B$ of an infinite group $G$ satisfying $|A| < |G|$, $|B| < |G|$ are separated? This is so if $A, B$ are finite, or $G$ is Abelian.