---

Claim. Alex Cohen, Cosmin Pohoata and Dmitrii Zakharov, Lower bounds for incidences, Invent. Math. 240 (2025), no. 3, 1045–1118, published online 14 March 2025; posted as arXiv:2409.07658 on 11 September 2024; carded at cohen_2024_lower_bounds_incidences. The paper proves lower bounds for incidences between points of the unit square and $\delta$-tubes, one tube through each point, under regularity conditions. Its consequence for Heilbronn's triangle problem, Theorem 1.8 in the numbering the release preprint on OpenAI's claim page cites, states that for every $\varepsilon>0$ and all large $n$, any $n$ points in the unit square contain three points forming a triangle of area at most $n^{-7/6+\varepsilon}$; that is, $\Delta(n)\le n^{-7/6+o(1)}$ for the square's quantity $\Delta(n)$. The route is the incidence theorem for points and tubes (Theorem 1.1): given a line through each of $n$ points of the unit square, some point lies within $n^{-2/3+o(1)}$ of another point's line (Corollary 1.2), and with the lines drawn through nearest neighbors this gives a triangle of area $n^{-7/6+o(1)}$. The bound improves the authors' earlier exponent $8/7+1/2000$ (arXiv:2305.18253) and the exponent $8/7$ of Komlós, Pintz and Szemerédi on their claim page.

Covers. The upper bound $\alpha(n)\ll n^{-7/6+o(1)}$ for the quantity of Problem 507: the disk of radius one lies in a square of side two, which scales to the unit square with every area divided by four, so $\alpha(n)\le4\Delta(n)\le4n^{-7/6+o(1)}$ (the transfer is a remark of this page). No lower bound, and not the order of $\alpha(n)$, which the recorded bounds leave between the exponents $-2$ and $-7/6$.

Depends on. No page of this wiki.

Acceptance. Refereed: Inventiones Mathematicae published the paper. The site's curator credits it with the upper bound in the problem's commentary, but the site labels the problem OPEN, so that credit is context and not reviewed evidence.