Problem 117

Erdős Problem #117: declared status 'open'. Formalized: no. Let $h(n)$ be minimal such that any group $G$ with the property that any subset of $>n$ elements contains some $x\neq y$ such that $xy=yx$ can be covered by at most $h(n)$ many Abelian subgroups. Estimate $h(n)$ as well as possible. Current best: Erd\H{o}s [Er97f] writes that the lower bound was already known to Isaacs. Prize: no. Tags: group theory.