OpenAI's Reasoning Model Disproves an 80-Year-Old Erdős Conjecture

For 80 years, one of the most famous unsolved problems in combinatorial geometry sat untouched. Mathematicians knew roughly how many pairs of points could sit exactly one unit apart in a flat plane, and most believed Paul Erdős had essentially nailed the ceiling back in 1946. Then an OpenAI reasoning model proved him wrong.

The planar unit distance problem, first posed by Paul Erdős in 1946, is one of the best-known questions in combinatorial geometry: easy to state and remarkably difficult to resolve. The question is deceptively simple: place n points in a 2D plane. How many pairs of those points can be exactly distance 1 apart? Since Erdős's original work, the prevailing belief was that square grid constructions were essentially optimal for maximizing the number of unit-distance pairs. An internal OpenAI model has now disproved this longstanding conjecture, providing an infinite family of examples that yield a polynomial improvement.

What Erdős actually conjectured

Erdős's conjecture was that the actual optimum was much closer to the lower bound than the upper one. He predicted, but couldn't prove, that the maximum number of unit-distance pairs grows just barely faster than the number of points. In math notation, he conjectured the maximum was n^(1+o(1)), where the extra term in the exponent shrinks toward zero as n grows. The best upper bound, O(n^(4/3)), dates to 1984 and had not meaningfully improved since.

The OpenAI model disproved this longstanding conjecture, discovering an infinite family of constructions using deep algebraic number theory , specifically Golod-Shafarevich theory and infinite class field towers , that achieve a polynomial improvement over square grids, specifically n^(1+δ) unit-distance pairs for some fixed δ > 0. The improvement has been quantified with an exponent of approximately 0.014, a figure later refined by Princeton mathematician Will Sawin.

A bridge from number theory to geometry

The most striking thing about the proof is not just that it worked, but how. Rather than iterating on known grid arrangements, the model approached the problem through algebraic number theory, connecting it to advanced mathematical structures called infinite class field towers. The result is an infinite family of configurations that surpass the traditionally accepted optimal ones, refuting Erdős's conjectured upper bound outright.