---

Claim. Write $\omega(G)$ for the largest size of a set of pairwise non-commuting elements of a group $G$, $a(G)$ for the least number of Abelian subgroups whose union is $G$, and $h(n)=\sup\{a(G):\omega(G)\leq n\}$, which is the function of Problem 117. Lecomte's Theorem 2.2 states that

$$\log_2 h(n)=\frac{n}{2}+O\!\left(\sqrt{n}\,(\log(n+2))^3\right),$$

so that $h(n)^{1/n}\to\sqrt{2}$. Pyber [Py87] had shown $c_1^n<h(n)<c_2^n$ for absolute constants $c_2>c_1>1$, with the lower bound known to Isaacs; the claim determines the base of the exponential. The manuscript Sharp Asymptotics for Abelian Covers of Groups with Bounded Noncommutativity by Guillaume Lecomte (independent researcher) was deposited on Zenodo on 18 August 2026 and claimed on the site's proof-claims tab the same day as a full proof; the same day the author announced the result in the problem's discussion thread, stating the rate above, saying that the problem is solved at the level Erdős posed it, and linking an intermediate Zenodo version. The record was revised five times; this page describes the sixth and latest version (record 22033543, dated 20 August 2026, 19 pages). The claim's tab names the AI system Fable; the manuscript's own statement says that generative AI tools were used as auxiliary tools for proof checking, consistency checks and TeX editing, with the mathematics remaining the author's responsibility. The claim value is answered because the problem asks for an estimate of $h(n)$ and the result is an asymptotic, neither a proof nor a disproof of a stated assertion.

Submission note. Posted to erdosproblems.com as a proof claim by Guillaume Lecomte (account guillaumelecomte) on 18 August 2026, giving "Fable" as the AI used:

I claim that the extremal number h(n) of abelian subgroups needed to cover a group with no pairwise noncommuting set larger than n satisfies log₂ h(n) = n/2 + o(n), so its exponential growth rate is √2. The lower bound comes from extraspecial 2-groups. For the upper bound, I reduce to finite groups and analyse p-groups through central factors of the derived subgroup, encoding commutation by alternating bilinear forms. Isotropic subspaces provide abelian covers, while large ranks force large noncommuting sets. The main difficulty is controlling interactions between successive levels; weak interactions are handled by a nested construction and strong ones by exact centralization. This gives the sharp coefficient 1/2 for 2-groups and a smaller coefficient for odd primes. Sylow decomposition and the Fitting subgroup then extend the estimate to general finite groups at polynomial cost. Fable was used for proof assistance, consistency checks, and LaTeX editing. Notes: A preprint of the full proof is available on Zenodo. I would be grateful for comments, corrections, or references to related work that I may have missed.

The argument. Both parameters depend only on the commutator structure modulo the center, so isoclinism (Lemma 2.1) reduces the problem to finite groups. Extraspecial $2$-groups give the lower bound. For the upper bound, a central series of the derived subgroup of a finite $p$-group $P$ yields alternating bilinear forms whose ranks measure the cost of a recursive Abelian cover: isotropic subspaces give covers, large ranks give large non-commuting sets. A nested construction absorbs weak interaction between the levels, and centralizing exactly deals with strong interaction, which gives the coefficient $1/2$ for $p=2$ and a smaller coefficient for odd primes (Theorem 6.1); Sylow decomposition extends the bound to nilpotent groups (Theorem 7.1). For a general finite group $G$ the proof passes to the nilpotent normal subgroup $H=C_G(G')$, whose index is quasipolynomially bounded through Pyber's conjugacy-class lemma and the Neumann and Vaughan-Lee estimate on the derived subgroup (Theorem 8.2), and a domination argument on cosets of $H$ finishes. Two corollaries record that the least possible index of an Abelian subgroup has the same exponential rate and that asymptotic extremality is concentrated in $2$-groups. In the thread the author reported on 2026-08-19 that a revision cites rather than reproves an earlier result at Lemma 3.1 and reworks the general finite reduction so that the main theorem no longer relies on the classification of finite simple groups.

Standing. Claimed. The site labels the problem OPEN (page last edited 23 January 2026) and marks proof claims as unexamined by anyone associated with it. The comments on the claim concern the AI-use disclosure and report an AI-assisted reading of the manuscript, which is not a review. The manuscript is also posted on arXiv (2608.20507v1, dated 20 August 2026), whose comment field records submission to Graphs and Combinatorics; a submission is not a publication, and no journal publication is recorded.