Problem 507

Let $\alpha(n)$ be such that every set of $n$ points in the unit disk contains three points which determine a triangle of area at most $\alpha(n)$ . Estimate $\alpha(n)$ .

Source: erdosproblems.com/507

Geometry

Wiki page

Status

Open

No claim settles this problem.

Open. The site labels the problem OPEN (page last edited 30 December 2025); its proof-claims thread carried no claim as of 6 October 2026. The claim pages record the refereed bounds of Komlós, Pintz and Szemerédi and of Cohen, Pohoata and Zakharov as accepted partial claims, a release result on the lower bound, pending, and two earlier arXiv preprints asserting stronger bounds for the disk, one conditional on an unproved assumption and one rejected.

CPZ23: Cohen, A. and Pohoata, C. and Zakharov, D., A new upper bound for the Heilbronn triangle problem. arXiv:2305.18253 (2023).
CPZ24: Cohen, A. and Pohoata, C. and Zakharov, D., Lower bounds for incidences. arXiv:2409.07658 (2024); Invent. Math. 240 (2025), no. 3, 1045-1118.
KPS81: Komlós, János and Pintz, János and Szemerédi, Endre, On Heilbronn's triangle problem. J. London Math. Soc. (2) 24 (1981), no. 3, 385-396.
KPS82: Komlós, János and Pintz, János and Szemerédi, Endre, A lower bound for Heilbronn's problem. J. London Math. Soc. (2) 25 (1982), no. 1, 13-24.
OAI26: OpenAI, A power improvement in the Heilbronn triangle lower bound. OpenAI Math Release preprint, 25 September 2026 (pinned PDF).

Problem 507

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Proof claims

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