Definition
The planar unit distance problem asks: given n points in the plane, what is the maximum number of pairs of points exactly distance 1 apart? Let u(n) denote this maximum. Paul Erdős posed the problem in 1946; it is among the best-known questions in combinatorial geometry.
Erdős Conjecture (Disproved May 2026)
Erdős conjectured u(n) ≤ n^(1+o(1)) — that square-grid constructions were essentially optimal. An internal openai reasoning model disproved this, constructing point sets with at least n^(1+δ) unit-distance pairs for fixed δ > 0.
Will Sawin (Princeton) refined δ to 0.014 in 2026-05-31-openai-erdos-sawin-explicit-bound. Best prior lower bound since Erdős (1946) was n^(1 + C/log log n). Upper bound O(n^(4/3)) unchanged since Spencer-Szemerédi-Trotter (1984).
Key Techniques
The disproof uses algebraic number theory — infinite class field towers, Golod–Shafarevich theory — applied to discrete geometry, an unexpected cross-domain connection.