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Claim. Every non-averaging $A\subseteq\{1,\ldots,N\}$ has $|A|\le N^{1/4+o(1)}$, and in particular the largest such set has size $N^{1/4+o(1)}$ (Theorem 1, p. 2 of arXiv v2). In the notation of Problem 186 this is $F(N)\le N^{1/4+o(1)}$ and, with Bosznay's lower bound, $F(N)=N^{1/4+o(1)}$: the order of growth the problem asks for is determined up to the $o(1)$ in the exponent, which is what the site's SOLVED label records. H. T. Pham and D. Zakharov, Sharp bound for the Erdős--Straus non-averaging set problem, Geom. Funct. Anal. 35 (2025), no. 6, 1712--1738, arXiv:2410.14624 (v1 18 October 2024, v2 10 September 2025), cited as [PhZa24] on the problem page. Library home pham_2024_sharp_bound_erdos_straus_non_averaging; result page Theorem 1. The paper's non-averaging condition, no element the average of a nonempty subset not containing it, is the problem's, since a one-element subset averages to itself. The route, as the introduction describes it: the subset-sums structure theorem of Conlon, Fox and Pham places a large set, after removing few elements, in a generalized arithmetic progression of bounded dimension whose multiple is filled by the subset sums, and a structural result on point sets in nearly convex position turns the non-averaging condition into a convexity constraint. The theorem improves the bound $N^{\sqrt2-1+o(1)}$ of Conlon, Fox and Pham, the first polynomial improvement on the Erdős--Sárközy bound $(N\log N)^{1/2}$. What remains open is recorded on the problem page: the $o(1)$, any constant, and the sequence $F(N)$ beyond the OEIS terms.

Depends on. Bosznay's lower bound $F(N)\gg N^{1/4}$, which the paper recalls on p. 1 as the best known lower bound and which the "in particular" clause of Theorem 1 consumes; the upper bound itself rests on the literature the paper cites, not on a page of this wiki.

Acceptance. Refereed: the paper appeared in Geometric and Functional Analysis, published online 3 December 2025 (per its Crossref record, as the problem page records). Reviewed: the site's curator, Thomas Bloom, credits the upper bound to Pham and Zakharov in the problem page's commentary and labels the problem SOLVED (page last edited 8 April 2026); that is documented acceptance outside this project.

Read depth. The library card checks the definition and Theorem 1; the proof (Sections 2--4) was not read. Nothing here rests on a review by this project.