Heilbronn's Triangle Problem

Heilbronn triangle problem: place \(s\) points in the unit disc so as to maximise the area of the smallest triangle they determine. What is the exact asymptotic order of this optimum \(\Delta(s)\)? (Known: \(\log s/s^2 \ll \Delta(s) \ll n^{-7/6+o(1)}\); Heilbronn's original \(O(1/s^2)\) guess is false.)

Heilbronn asked it in the 1940s; his conjectured \(O(1/s^2)\) rate was refuted by Komlós–Pintz–Szemerédi (1982), who built configurations with minimal-triangle area \(\gg \log s / s^2\). Roth (1951) gave the first nontrivial upper bound; the record has seesawed since, with Cohen–Pohoata–Zakharov (2023) reaching \(n^{-7/6+o(1)}\) from above. The exact order of the optimal spread remains unknown — a rare classical geometry problem where upper and lower bounds refuse to meet.