An OpenAI model has disproved a central conjecture in discrete geometry

For nearly 80 years, mathematicians have studied a deceptively simple question: if you place nn points in the plane, how many pairs of points can be exactly distance 11 apart?

This is the planar unit distance problem, first posed by Paul Erdős in 1946. It is one of the best-known questions in combinatorial geometry, easy to state and remarkably difficult to resolve. The 2005 book Research Problems in Discrete Geometry, by Brass, Moser, and Pach, calls it “possibly the best known (and simplest to explain) problem in combinatorial geometry.” Noga Alon, a leading combinatorialist at Princeton, describes it as “one of Erdős’ favorite problems.” Erdős even offered a monetary prize for resolving this problem.

Today, we share a breakthrough on the unit distance problem. Since Erdős’s original work, the prevailing belief has been that the “square grid” constructions depicted further below were essentially optimal for maximizing the number of unit-distance pairs. An internal OpenAI model has disproved this longstanding conjecture, providing an infinite family of examples that yield a polynomial improvement. The proof has been checked by a group of external mathematicians. They have also written a companion paper explaining the argument and providing further background and context for the significance of the result.

The result is also notable for how it was found. The proof came from a new general-purpose reasoning model, rather than from a system trained specifically for mathematics, scaffolded to search through proof strategies, or targeted at the unit distance problem in particular. As part of a broader effort to test whether advanced models can contribute to frontier research, we evaluated it on a collection of Erdős problems. In this case, it produced a proof resolving the open problem.

Mathematicians on the result

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Previously known construction of many unit distances from a rescaled square grid.

nn, the proof constructs configurations of nn points with at least n1+δn^{1+\delta} unit-distance pairs, for some fixed exponent δ>0\delta > 0. (The original AI proof does not give an explicit δ\delta, but a forthcoming refinement due to Princeton mathematics professor Will Sawin has shown one can take δ=0.014\delta=0.014.)O(n4/3)O(n^{4/3}), dates to work by Spencer, Szemerédi, and Trotter in 1984, and despite later refinements and related structural work by Székely, Katz and Silier, Pach, Raz, and Solymosi and by others, the upper bound has remained essentially unchanged. As evidence in favor of the conjecture, Matoušek and Alon-Bucić-Sauermann studied the problem with non-Euclidean distances in the plane, and proved that "most" of these non-Euclidean distances obey the conjecture in some sense.