Problem 507

Erdős Problem #507: declared status 'open'. Formalized: yes. 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 $\alpha(n)$. Current best: 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 Koml\'{o}s, Pintz, and Szemer\'{e}di [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 Koml\'{o}s, Pintz, and Szemer\'{e}di [KPS81]). and Zakharov, D., A new upper bound for the Heilbronn triangle problem. [KPS82] Koml\'{o}s, J\'{a}nos and Pintz, J\'{a}nos and Szemer\'{e}di, Endre, A lower bound for Heilbronn's problem. Prize: no. Tags: geometry.