Problem Statement
Let $\alpha(n)$ be such that every set of $n$ points in the unit disk contains three points which determine a triangle of area at most $\alpha(n)$. Estimate $\alpha(n)$.
Categories:
Geometry
Progress
Heilbronn's triangle problem. It is trivial that $\alpha(n) \ll 1/n$. Erdős observed that $\alpha(n)\gg 1/n^2$. The current best bounds are\[\frac{\log n}{n^2}\ll \alpha(n) \ll \frac{1}{n^{7/6+o(1)}}.\]The lower bound is due to Komlós, Pintz, and Szemerédi [KPS82]. The upper bound is due to Cohen, Pohoata, and Zakharov [CPZ24] (improving on their earlier work [CPZ23] which itself improves an exponent of $8/7$ due to Komlós, Pintz, and Szemerédi [KPS81]).This problem is Problem 77 on Green's open problems list.
Source: erdosproblems.com/507 | Last verified: January 15, 2026