Place \(n\) points in the plane. What is the maximum possible number of pairs at distance exactly \(1\)? Erdős (1946) achieved slightly superlinear growth \(n^{1+c/\log\log n}\) via carefully scaled square grids and conjectured this essentially optimal: \(u(n) < n^{1+\varepsilon}\) for every \(\varepsilon > 0\). Best known upper bound: \(O(n^{4/3})\) (Spencer–Szemerédi–Trotter 1984).
"Possibly the best known and simplest to explain problem in combinatorial geometry" (Brass–Moser–Pach), and one of Erdős's own favourites, with money on it from his prize list. Eighty years of attempts assumed the grid was unbeatable.
Disproved, autonomously, by an OpenAI reasoning model — announced 20 May 2026. The model produced an infinite family of configurations of n points with at least \(n^{1+δ}\) unit-distance pairs for a fixed δ > 0, refuting the \(n^{1+o(1)}\) belief. The construction replaces Erdős's Gaussian-integer grids with richer rings of algebraic integers carrying more symmetries — ideas imported wholesale from algebraic number theory into elementary geometry. Will Sawin (Princeton) refined the argument to an explicit δ = 0.014.
External verification was led by Tim Gowers, Noga Alon, Melanie Matchett Wood, Thomas Bloom, Daniel Litt and others; Gowers called it "a milestone in AI mathematics". Notably, the model solves the problem in only about half of repeated attempts — persistence, not magic. The problem itself stays open: the truth lies somewhere above \(n^{1.014}\) and below \(n^{4/3}\).
Context: the companion distinct-distances problem fell to near-sharp bounds via Guth–Katz (2015), leaving unit distances as the hard direction — typical Erdős asymmetry. See the distinct-distances entry here. The algebraic-integer machinery (rings with large unit groups supplying symmetries) is now the method to watch across discrete geometry.
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