A power improvement in the Heilbronn triangle lower bound

Some n points in a unit square keep every triangle above n^(−2+η) in area, beating 1/n² by a fixed power.

1 paper September 25, 2026 Explained by AI

Heilbronn's problem asks how to place n points in a square so the smallest triangle they form is as large as possible. Simple constructions give about 1/n², and a 1982 result gained a logarithm. Many expected that no fixed power of n could be gained. This paper shows that one can be gained, though the power is extremely small.

First step

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Significance

Major

It breaks a widely discussed form of Heilbronn's problem by a power of n, though a tiny one.

Trust

Formal proof listed

Field

Combinatorics

1 paper

The explanation

Put four points at the corners of a square with sides of length 1. Any three of them make a triangle of area 1/2. Now add a fifth point at the centre. Two opposite corners and the centre lie on one line. Their triangle is flat, with area 0. So adding one point in the obvious place ruined everything.

Heilbronn's triangle problem asks how to place n points in a unit square to keep every triangle large. Look at all triangles the points form, and take the smallest area. Call the best possible value Δ(n). The question is how fast Δ(n) shrinks as n grows.

The original guess was that Δ(n) shrinks at least as fast as a constant times 1/n²Source 1. For 100 points, 1/n² is 0.0001. Simple constructions reach that scale, including one by Erdős based on a parabolaSource 2. In 1982, Komlós, Pintz and Szemerédi beat it by a factor of log n. That disproved the original guessSource 3. A weaker guess survived. It says Δ(n) is at most about n^(−2+ε), for every small ε, once n is largeSource 4. In words, nobody can beat 1/n² by a fixed power of n.

This paper proves that weaker guess falseSource 5. It claims more: for every large n, some n points have all triangles of area at least c·n^(−2+η). Here c and η are fixed positive numbers. The gain η is a real power of n, not a logarithm. But the paper says the explicit η it can give is extremely smallSource 6. So the improvement matters in principle, not at any size one could draw.

From the other side, upper limits have also improved. In 1951, Roth proved the first upper bound that beats 1/nSource 7. Cohen, Pohoata and Zakharov later showed Δ(n) is at most about n^(−7/6)Source 8. The true rate lies somewhere between these lower and upper bounds.

The difficulty is that random points form many thin triangles. Scattering points at random and deleting thin triangles only reaches the 1/n² scaleSource 2. The paper starts from Erdős's parabola over a finite number system, where no three distinct labelled points line upSource 9. It lifts these to whole-number coordinates, using careful digit expansions to keep triangle areas away from zero. Only rare bad triples remain, and these are deleted.

The paper notes that earlier preprints claimed stronger lower boundsSource 10. It points to specific gaps in their arguments. It does not claim those bounds are false.

How sure is this? A computer proof checker called Lean has checked part of the claim. The Lean scope note says it builds point sets for an unbounded sequence of sizes n, with every triangle at least n^(−2+η) in area. That is enough to refute the weaker guess. The paper's statement for every large n is broader and lies outside that check. No specialist review of the paper has been published.

What it doesn't show. This does not give a usable exponent; the gain is real in principle but extremely small.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major if it holds, with a tiny exponent. Lean has checked a version for an unbounded sequence of n; the every-large-n statement and specialist review are missing.

Words used

Heilbronn's triangle problem: How to place n points in a unit square so the smallest triangle they form is as large as possible.
Δ(n): The largest possible smallest-triangle area for n points in the unit square.
Collinear: Lying on one straight line, which makes a triangle with zero area.
Finite field: A number system with finitely many values where adding, multiplying and dividing all work.
Power improvement: A gain by a factor n^η for a fixed η > 0, rather than a slowly growing factor like log n.

The original paper

A power improvement in the Heilbronn triangle lower bound (PDF, opens in a new tab)
September 25, 2026

Lean notes

All quoted lines (10)

The original Heilbronn conjecture asked whether $\Delta(n)=O(n^{-2})$ ; see Roth.
A power improvement in the Heilbronn triangle lower bound Back to text
Two elementary constructions attain this scale.
A power improvement in the Heilbronn triangle lower bound Back to text
Koml\'os, Pintz and Szemer\'edi improved this to $\Delta(n)\gg(\log n)/n^2$ , disproving the original conjecture.
A power improvement in the Heilbronn triangle lower bound Back to text
The almost $n^{-2}$ formulation, discussed in Zakharov's survey, asks whether, for every $\varepsilon>0$ , there are constants $C_\varepsilon$ and $n_0(\varepsilon)$ such that $\Delta(n)\le C_\varepsilon n^{-2+\varepsilon} \qquad(n\ge n_0(\varepsilon)).$
A power improvement in the Heilbronn triangle lower bound Back to text
We disprove this formulation by a power improvement in the lower bound.
A power improvement in the Heilbronn triangle lower bound Back to text
Section … gives an explicit, though extremely small, admissible exponent.
A power improvement in the Heilbronn triangle lower bound Back to text
On the upper-bound side, Roth proved the first $o(n^{-1})$ estimate by a density-increment argument, followed by refinements of Schmidt and Roth.
A power improvement in the Heilbronn triangle lower bound Back to text
Cohen, Pohoata and Zakharov developed new incidence-geometric approaches, first obtaining $\Delta(n)\le n^{-8/7-1/2000}$ for sufficiently large $n$ , and subsequently proving $\Delta(n)\ll_\varepsilon n^{-7/6+\varepsilon}$ for every $\varepsilon>0$ .
A power improvement in the Heilbronn triangle lower bound Back to text
The columns $(1,\xi,\xi^2)^{\mathsf T}$ have nonzero determinant whenever their labels are distinct.
A power improvement in the Heilbronn triangle lower bound Back to text
Earlier preprints claim stronger power lower bounds than ours, with unresolved issues in the following versions.
A power improvement in the Heilbronn triangle lower bound Back to text

Other sources

openai/math, Lean scope note for family 191 (2026) The linked construction is for an unbounded sequence of sizes; the paper's statement for every sufficiently large $n$ is broader.
openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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