Problem 117

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.

Source: erdosproblems.com/117

Group theory

Wiki page

Status

ClaimedAnswered

A full solution has been claimed but not yet accepted. Settled in another form, for example when its parts resolve differently or the question is open-ended.

Open on the site (label OPEN; page last edited 23 January 2026). The site's proof-claims tab carries one full proof claim, submitted 18 August 2026 by Guillaume Lecomte and recorded as a pending claim on its claim page (Lecomte, 2026): that $\log_2 h(n)=n/2+o(n)$ , so $h(n)^{1/n}\to\sqrt{2}$ .

Problem 117

Status

Proof claims

Discussion