Cohen 2023 new upper bound heilbronn triangle problem

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Alex Cohen, Cosmin Pohoata, Dmitrii Zakharov, A new upper bound for the Heilbronn triangle problem. arXiv:2305.18253 (2023).

Theorem 1.1 proves that, for every sufficiently large n, among any n points in the unit square some three span a triangle of area at most n^{-8/7-1/2000}, the first polynomial improvement over the bound exp(c sqrt(log n)) n^{-8/7} of Komlós, Pintz and Szemerédi from 1981 (dated 1982 in the paper's abstract). Theorem 1.2 gives a simple proof of the stronger bound Delta <= n^{-7/6+eps}, for every eps > 0 and large n, for homogeneous point sets (at most C points in every axis-parallel n^{-1/2} by n^{-1/2} square). The approach builds new connections between the Heilbronn problem and incidence geometry and projection theory tied to the discretized sum-product phenomenon, with a key input a theorem of Orponen, Shmerkin and Wang; the paper also reproves the Komlós-Pintz-Szemerédi result as an intermediate step. The authors note the exponent 1/2000 is not optimized and arises from a system of inequalities. The paper bears on problem 507 by giving the then-record upper bound for the Heilbronn triangle function Delta(n) of the unit square, whose known lower bound is Omega(log n / n^2); problem 507 asks for the unit-disk quantity alpha(n), and since the unit disk lies in a square of side two, which scales to the unit square with areas divided by four, alpha(n) <= 4 Delta(n) carries the bound over.

Source: https://arxiv.org/abs/2305.18253. The arXiv record (https://arxiv.org/abs/2305.18253, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Bears on. #507

Results to transcribe.

• Theorem 1.1: For sufficiently large n, any n points in [0,1]^2 contain three forming a triangle of area at most n^{-8/7-1/2000}.
• Theorem 1.2: For any ε>0 and n large, any homogeneous set of n points in [0,1]^2 contains a triangle of area at most n^{-7/6+ε}.