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Claim. J. Komlós, J. Pintz and E. Szemerédi, A lower bound for Heilbronn's problem, J. London Math. Soc. (2) 25 (1982), no. 1, 13–24. Write $\Delta(n)$ for the largest $a$ such that some $n$ points of the unit square have every triangle with vertices among them of area at least $a$. The paper proves that there is an absolute constant $c>0$ with $\Delta(n)\ge c\,(\log n)/n^2$ for all sufficiently large $n$: for every large $n$ there are $n$ points in the unit square with every triangle of area at least $c(\log n)/n^2$. This refutes Heilbronn's conjecture that $\Delta(n)=O(n^{-2})$ and improves Erdős's lower bound of order $n^{-2}$ by the factor $\log n$. The proof is probabilistic: it places many random points, forms the hypergraph of their small triangles, and takes a large independent set in it, using a theorem on independent sets in sparse three-uniform hypergraphs with few short cycles. The paper has no library card.

Covers. The lower bound $\alpha(n)\gg(\log n)/n^2$ for the quantity of Problem 507: a translate of the unit square lies in the disk of radius one and translation preserves areas, so $\alpha(n)\ge\Delta(n)\ge c(\log n)/n^2$ for all large $n$ (the transfer is a remark of this page). No upper bound, and not the order of $\alpha(n)$; the bound is the best published lower bound, with the pending power improvement on OpenAI's claim page.

Depends on. No page of this wiki.

Acceptance. Refereed: the Journal of the London Mathematical Society published the paper. The site's curator credits it with the lower bound in the problem's commentary, but the site labels the problem OPEN, so that credit is context and not reviewed evidence.