Result 191, Combinatorics

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+cn^{-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.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

How large can the smallest triangle be when n points are placed in a unit square? The manuscript reports configurations that surpass a conjectured limit, changing what geometry can force in crowded point sets.

What changes?

Heilbronn's triangle problem asks how to arrange n points in the unit square to maximize the area of the smallest triangle formed by any three points. The manuscript claims that, for every sufficiently large integer n, an arrangement exists in which every triangle has area at least c1 times n raised to the power negative two plus eta. Both c1 and eta are positive constants independent of n; eta is fixed but extremely small.

What does that help mathematicians do?

Relative to the inverse-square scale, the claimed guarantee gains a factor that grows as a fixed positive power of n. However small that power is, it eventually exceeds corrections growing more slowly than every positive power. This rules out the conjectured almost inverse-square universal upper bound. Researchers must therefore allow larger minimum triangle areas when seeking a sharp bound, although this result does not identify the optimal scale.

Are there practical applications?

The immediate value is foundational: the result changes the limits that extremal geometry seeks to establish for planar point arrangements. It supplies configurations against which proposed universal guarantees of small triangles must be checked. The supplied abstract gives no computational efficiency or implementation guarantees, so the construction should not be presented as a practical point-placement algorithm.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A power improvement in the Heilbronn triangle lower bound

September 25, 2026 24 pages

There are absolute constants η,c1>0\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 c1n−2+ηc_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.

PDF LaTeX source
Cite (BibTeX) 
@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's note on what the formalization covers, from lean/docs/191.md.

A power improvement in the Heilbronn triangle lower bound

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

Heilbronn's triangle problem asks how large the smallest triangle determined by nn points in the unit square can be. The formalization constructs an unbounded sequence of sizes nn and point sets for which every triangle has area at least n−2+ηn^{-2+\eta}, for one fixed η>0\eta>0. It consequently refutes the proposed upper bound of order n−2+εn^{-2+\varepsilon} for every ε>0\varepsilon>0. The linked construction is for an unbounded sequence of sizes; the paper's statement for every sufficiently large nn is broader.

Comparator links

Result Comparator statement
Power improvement for the smallest Heilbronn triangle HeilbronnTriangle.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.