OP-01140 Solution claimed

Heilbronn's triangle problem

Let $\alpha(n)$ be the largest $\alpha$ such that some $n$ points of the unit disc determine no triangle of area less than $\alpha$. Komlós, Pintz and Szemerédi proved $\alpha(n) \gg n^{-2} \log n$, so that the $o(1)$ below cannot be dropped, and $\alpha(n) \le n^{-8/7 + o(1)}$. Cohen, Pohoata and Zakharov improved the upper bound to $n^{-7/6 + o(1)}$.

Problem. Is $\alpha(n) \le n^{-2 + o(1)}$?

Source: H. A. Heilbronn; see J. Komlós, J. Pintz and E. Szemerédi (1981, 1982) and Erdős problem #507. Listed as Problem 77 in B. Green, 100 open problems (version of January 2026).

Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.

Classification

Primary: CO — Combinatorics
Secondary: MG
Keywords: discrete geometry, Erdős problems, Heilbronn triangle problem
Literature checked: Nov. 3, 2026 by clemens-admin

References

@article{kps1981heilbronn,
  author = {Koml{\'o}s, J. and Pintz, J. and Szemer{\'e}di, E.},
  title = {On {H}eilbronn's triangle problem},
  journal = {Journal of the London Mathematical Society (2)},
  volume = {24},
  number = {3},
  pages = {385--396},
  year = {1981},
  doi = {10.1112/jlms/s2-24.3.385}
}

@article{kps1982heilbronn,
  author = {Koml{\'o}s, J. and Pintz, J. and Szemer{\'e}di, E.},
  title = {A lower bound for {H}eilbronn's problem},
  journal = {Journal of the London Mathematical Society (2)},
  volume = {25},
  number = {1},
  pages = {13--24},
  year = {1982},
  doi = {10.1112/jlms/s2-25.1.13}
}

@misc{cpz2023heilbronn,
  author = {Cohen, Alex and Pohoata, Cosmin and Zakharov, Dmitrii},
  title = {A new upper bound for the {H}eilbronn triangle problem},
  year = {2023},
  eprint = {2305.18253},
  archivePrefix = {arXiv}
}

@article{cpz2025incidences,
  author = {Cohen, Alex and Pohoata, Cosmin and Zakharov, Dmitrii},
  title = {Lower bounds for incidences},
  journal = {Inventiones Mathematicae},
  volume = {240},
  number = {3},
  pages = {1045--1118},
  year = {2025},
  eprint = {2409.07658},
  archivePrefix = {arXiv}
}

@misc{openai2026heilbronn,
  author = {{OpenAI}},
  title = {A power improvement in the {H}eilbronn triangle lower bound},
  howpublished = {\url{https://github.com/openai/math/blob/4eaee87548250b2682affda7d05d947c05554bae/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/main.pdf}},
  note = {AI-generated, unrefereed; dated 25 September 2026},
  year = {2026}
}

@misc{green100open,
  author = {Green, Ben},
  title = {100 open problems},
  howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
  note = {Problem 77; version of January 2026, most recent update December 2025},
  year = {2026}
}

Resolution claims

A power improvement in the Heilbronn triangle lower bound v1 GitHub ↗

OpenAI

There are absolute constants $\eta,c_1>0$ such that, for every sufficiently large integer $n$, one can choose $n$ points in the unit square so that every triangle they determine has area at least $c_1n^{-2+\eta}$. Thus the almost $n^{-2}$ upper-bound formulation of Heilbronn's triangle problem is false. The exponent $\eta$ is fixed but extremely small.

Counterexample · not verified · fidelity not established · posted on GitHub 6 Oct 2026

Status history

When	Change	By	Reason
2026-11-08	Needs verification → Solution claimed	—	Recorded here: math @ 4eaee87, build/main.tex.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-01140, "Heilbronn's triangle problem", clopen.solutions.