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Claim. For some $c>0$ and all sufficiently large $N$, the largest non-averaging subset of $\{1,\ldots,N\}$ has more than $cN^{1/4}$ elements: $F(N)\gg N^{1/4}$ in the notation of Problem 186. The witness is the set of the $q-1$ integers $iq^3+i(i+1)/2$, $1\le i\le q-1$, all below $q^4$: reading each in base $q^2$ places it at the point $(iq,\,i(i+1)/2)$ of a parabola, and a weighted average of distinct points of a strictly convex curve never lies on the curve, so no member is the mean of two or more others. Á. P. Bosznay, On the lower estimation of non-averaging sets, Acta Math. Hungar. 53 (1989), no. 1--2, 155--157, the paper's single theorem (printed p. 155) with its one-page proof; cited as [Bo89] on the problem page. Library home bosznay_1989_lower_estimation_non_averaging_sets; result page Theorem. The paper's $f(n)$ is the problem's $F(N)$: its non-averaging sets are those in which the mean of two or more members never belongs to the set, the problem's definition.

Covers. The lower bound $F(N)\gg N^{1/4}$ alone. The matching upper bound $F(N)\le N^{1/4+o(1)}$ is Pham and Zakharov's (claim page), and together they give the order of growth $F(N)=N^{1/4+o(1)}$ up to the $o(1)$ in the exponent; the constant and the $o(1)$ are not determined by either.

Depends on. No page of this wiki: the construction and its proof are self-contained.

Acceptance. Refereed: the paper is the publisher's version of record in Acta Mathematica Hungarica (its Crossref record dates the issue to March 1989 without a day, so this page is named by the first day of that month; the paper was received 14 August 1986). Reviewed: the site's curator, Thomas Bloom, credits the lower bound $N^{1/4}\ll F(N)$ to Bosznay in the problem page's commentary (label SOLVED, page last edited 8 April 2026), and Pham and Zakharov (p. 1 of arXiv v2) and Conlon, Fox and Pham (p. 4) each restate the construction as the best known lower bound; that is documented acceptance outside this project. The library card records the proof as read and followed in full, which is not an independent review.