---

Claim. J. Komlós, J. Pintz and E. Szemerédi, On Heilbronn's triangle problem, J. London Math. Soc. (2) 24 (1981), no. 3, 385–396. With $\Delta(n)$ the largest $a$ such that some $n$ points of the unit square have every triangle of area at least $a$, the paper proves that there is an absolute constant $c>0$ with

$$\Delta(n)\le\exp\bigl(c\sqrt{\log n}\bigr)\,n^{-8/7}$$

for all large $n$, so any $n$ points of the unit square contain three forming a triangle of area at most $n^{-8/7+o(1)}$. The exponent $8/7$ remained the best upper bound until Cohen, Pohoata and Zakharov lowered it, first to $8/7+1/2000$ and then to $7/6$. The paper has no library card.

Covers. The upper bound $\alpha(n)\le4\Delta(n)\ll n^{-8/7+o(1)}$ for the quantity of Problem 507, through the inclusion of the disk of radius one in a square of side two (a remark of this page). The bound is superseded by the exponent $7/6$ of Cohen, Pohoata and Zakharov on their claim page; it gives no lower bound and not the order of $\alpha(n)$.

Depends on. No page of this wiki.

Acceptance. Refereed: the Journal of the London Mathematical Society published the paper. The site's commentary names the exponent $8/7$ as the one the later bounds improved, on a problem the site labels OPEN, so that mention is context and not reviewed evidence.