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A power improvement in the Heilbronn triangle lower bound
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Abstract
There are absolute constants $\eta,c_1\gt 0$ 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 η is fixed but extremely small.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Ivan Blinov (@littleBro), heilbronn-triangle-bounds (GitHub repository, conditional research draft). Calls itself a conditional extension of this manuscript (pinned at commit fd4aeeb), using extension degree d = 5 and a shifted sphere packing to state a conditional exponent η = 1/145,278,430, with selected parts formalized in Lean 4 (not the full Heilbronn theorem); it says it was made with "substantial OpenAI Codex assistance" and has no peer review. [AI-generated per source]
Formalization
- mathcommunity/oai-math-revised, lean/docs/191.md (OpenAI Math Release Lean scope note, 2026-10-06). Describes a formalized power lower bound for an unbounded sequence of sizes $n$, sufficient to refute the almost $n^{-2}$ upper bound, and explicitly distinguishes this from the manuscript's assertion for every sufficiently large $n$. [family-level]
- OpenAI, lean/ComparatorChallenges/HeilbronnTriangle.lean (openai/math, commit adc7f12). Comparator statements
heilbronn_power_lower_boundandalmost_n_minus_two_refuted, with the explicit exponent 1/(100000·K), where K is built from binomial coefficients. - OpenAI, lean/OAI/Geometry/HeilbronnTriangle/Main.lean (openai/math, commit adc7f12). The Lean proofs of the two comparator theorems; they rely on imported modules (
Definitions,Refutation,ActualWeightedFiberCount).
Possibly relevant literature (found by the agent, not checked)
- C. Bertram-Kretzberg, T. Hofmeister, H. Lefmann, An Algorithm for Heilbronn's Problem (SIAM J. Comput. 30(2), 383–390, 2000). A deterministic polynomial-time version of the Komlós–Pintz–Szemerédi (log n)/n² lower bound, which is the bound the paper's Theorem 1.1 improves by a power.
- Tao Jiang, Ming Li and Paul Vitányi, The Average-Case Area of Heilbronn-Type Triangles (Random Structures & Algorithms 20(2), 2002; arXiv:math/9902043). Its order-$n^{-3}$ expected minimum triangle area for independent uniform points gives a probabilistic comparison point for the manuscript's arithmetic sampling and deletion construction behind the $n^{-2+\eta}$ lower bound.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 191
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026,
author = {{OpenAI}},
title = {{A power improvement in the Heilbronn triangle lower bound}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/main.pdf}{OAI:A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/191.md. The Lean statements have not been checked by people against the paper's claims.
Family 191: A power improvement in the Heilbronn triangle lower bound
For every sufficiently large n, constructs n points in the unit square such that every triangle has area at least $n^{-2+c}$ for one absolute c > 0. This disproves the conjectured almost-n−2 upper bound in Heilbronn's triangle problem, which asks how large the smallest determined triangle can be.
This is the only paper in this family.