Heilbronn's triangle problem: a power-saving lower bound n^(-2+eta), refuting the almost n^-2 upper bound

For $n$ points in the unit square let $\Delta(P)$ be the least area of a triangle they determine, and $\Delta(n)=\max_{|P|=n}\Delta(P)$ . Heilbronn asked for the order of $\Delta(n)$ , conjecturing $\Delta(n)=O(n^{-2})$ (reported by Roth, 1951); Erdos's parabola gives $\Delta(n)\gg n^{-2}$ . Komlos, Pintz and Szemeredi (1982) disproved the conjecture with $\Delta(n)\gg(\log n)/n^2$ , and the best upper bound is $n^{-7/6+o(1)}$ (Cohen, Pohoata and Zakharov). The remaining belief, the almost $n^{-2}$ formulation discussed in Zakharov's survey, is that $\Delta(n)\le C_\epsilon n^{-2+\epsilon}$ for every $\epsilon>0$ . What is the order of $\Delta(n)$ ; in particular, is $\Delta(n)=n^{-2+o(1)}$ ?

Result: Disproved(see note)
Status: Partial result
AI contribution: AI-discovered
Method: Construction
Field: Discrete geometry; Heilbronn's triangle problem
Posed by: Hans Heilbronn (reported by K. F. Roth, 1951); the almost n^-2 formulation as discussed in Zakharov's survey
Year posed: 1951
Years open: 75y
Solved: 2026-09-25
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 40 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: absolute constants $\eta,c_1>0$ and $n_0$ with $\Delta(n)\ge c_1n^{-2+\eta}$ for every $n\ge n_0$ . Hence the almost $n^{-2}$ upper-bound formulation is false. The construction samples integer columns in a box with finite-field norm congruence conditions and deletes the few small triangles; $\eta$ is explicit but extremely small. It does not determine the order of $\Delta(n)$ : the gap to the upper bound $n^{-7/6+o(1)}$ remains, and no conjecture for the true exponent is made.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The single manuscript (September 25, 2026) is the whole family.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Heilbronn's problem and the almost $n^{-2}$ formulation (1.1); it gives $\Delta(n)\ge c_1n^{-2+\eta}$ for all large $n$ , with an explicit but tiny $\eta$ . The proof was not refereed. The challenge lean/ComparatorChallenges/HeilbronnTriangle.json (solution module OAI.Geometry.HeilbronnTriangle.Main, present at the pinned commit) is not in the formalization catalogue; its statement HeilbronnTriangle.lean was read here. heilbronn_power_lower_bound gives an explicit $\eta=1/(10^5K)>0$ and point sets of sizes $n_j\to\infty$ in $[0,1]^2$ with every triangle of area at least $n_j^{-2+\eta}$ ; almost_n_minus_two_refuted states that $\Delta(n)\le Cn^{-2+\eta/2}$ fails for all large $n$ . This states the refutation; the 'every sufficiently large $n$ ' form is broader than the Lean. Not rebuilt here. The manuscript notes two earlier preprints (Ellmann, Agama) claiming stronger power bounds, with gaps it identifies.

Heilbronn's triangle problem: a power-saving lower bound n^(-2+eta), refuting the almost n^-2 upper bound

What was actually shown

What the AI did

Verification

Sources

Discussion