Erdős Unit Distance — How Many Unit-Distance Pairs Among $n$ Points?

For $n$ points in the plane, what is the maximum number $u(n)$ of pairs at exactly unit distance? Erdős (1946) showed $u(n) \ge n^{1 + c/\log\log n}$ via grid analysis and conjectured the upper bound matches. For 80 years the integer grid was believed to be essentially optimal.

A sister of the distinct-distances problem (both posed in the same 1946 Erdős paper). The integer grid was believed to be essentially optimal for 80 years, with the best general upper bound $O(n^{4/3})$ (Spencer–Szemerédi–Trotter, 1984). On 2026-05-20 OpenAI announced that an internal general-purpose reasoning model had automatically derived a counterexample refuting the grid-optimal intuition, verified by external mathematicians.

Lower bound: Erdős 1946 ($n^{1 + c/\log\log n}$, via grid + Gaussian-integer prime factorizations). Upper bound: Spencer–Szemerédi–Trotter 1984 ($O(n^{4/3})$). On 2026-05-20 OpenAI announced an automatically-derived counterexample to the grid-optimal intuition, verified externally with algebraic number theory tools. The main upper-bound conjecture itself remains open — only the methodological landscape has changed.