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Claim. Gabor Ellmann, A lower bound for Heilbronn's triangle-problem, arXiv:1703.03297, first posted 8 March 2017, withdrawn by version 3 of 18 March 2017 and reposted from version 4 of 5 March 2022; version 12 of 12 November 2025 is the one the release preprint of OpenAI's claim discusses. Its Theorem 1 asserts a construction of $n$ points in the unit circle, the disk of radius one of Problem 507, with every triangle of area at least $c\,n^{-3/2}(\log n)^{-7/2}$ for all large $n$, with a constant $c$ at most about $1.655$. That is the lower bound $\alpha(n)\gg n^{-3/2}(\log n)^{-7/2}$, a power stronger than the Komlós–Pintz–Szemerédi bound $(\log n)/n^2$ and than the release's $n^{-2+\eta}$. The construction places points on a system of concentric circles in a disk-ring domain, deletes points near the intersections of lines through pairs of points with the circles, and projects the result onto the unit circle.

Hypothesis. The claim is conditional on an assumption the manuscript does not prove: version 12 says that its method is heuristic and that it assumes the intersection points of the lines through pairs of points with each circle to be uniformly distributed along that circle, also between neighboring vertices, an assumption it says makes the article heuristic; the deletion estimate, its Theorem 4, uses it. The release preprint records the same point. So the result is recorded here as a conditional claim, with that local uniformity assumption as its hypothesis, and not as a proof of the bound.

Depends on. No page of this wiki.

Acceptance. None recorded. The preprint has no journal record, the site's page labels the problem OPEN with the Komlós–Pintz–Szemerédi bound as the best lower bound and does not mention it, and no outside review of it is recorded. The release says that its objection concerns the argument and not the possibility of the asserted bound.