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Claim. If $a\in A$ is the average of a nonempty $B\subseteq A\setminus\{a\}$ then $|B|\,a=\sum_{b\in B}b$, so $a$ divides a sum of distinct other elements of $A$; a non-dividing set is therefore non-averaging, and $F(N)\le h(N)$, where $h(N)$ is the largest size of a non-averaging subset of $\{1,\ldots,N\}$. Theorem 1 of Pham and Zakharov states that every non-averaging $A\subseteq[n]$ has $|A|\le n^{1/4+o(1)}$. Hence $F(N)\le N^{1/4+o(1)}$, which contradicts $F(N)>N^{1/2-o(1)}$ for all large $N$: the displayed question of Problem 131 is answered no. The paper does not mention the problem; the inclusion is the site's observation, checked in one line on the problem page. H. T. Pham and D. Zakharov, Sharp bound for the Erdős--Straus non-averaging set problem, arXiv:2410.14624 (v1 18 October 2024, the date this page is named by; v2 10 September 2025), Geom. Funct. Anal. 35 (2025), no. 6, 1712--1738, DOI 10.1007/s00039-025-00728-8 (the journal text not compared). The statement is paged as Theorem 1 of the library card, stated on the preprint's p. 2; the proof, which rests on the Conlon--Fox--Pham structure theorem for subset sums, is not examined on this page. The matching lower bound $h(n)\gg n^{1/4}$ is Bosznay's construction and is not part of this claim.

Covers. The displayed question only: $F(N)>N^{1/2-o(1)}$ is false, and $F(N)\le N^{1/4+o(1)}$. Not covered: the estimate of $F(N)$, to which the site's label attaches, open between the $N^{1/5}$ construction and this bound; the pending full claim on it is the Xeff claim page.

Acceptance. Refereed: the journal publication cited above (Geometric and Functional Analysis, December 2025). The site's curator, Thomas Bloom, records the inclusion, the bound and the negative answer to the displayed question in the commentary (page last edited 30 September 2025, accessed 2026-09-18 and 2026-10-07), but the site labels the problem OPEN, so the commentary is not acceptance and the claim lists no reviewed evidence. Nothing here is this project's own review beyond the one-line inclusion.

Depends on. No page of this wiki.