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An OpenAI model has disproved a central conjecture in discrete geometry | OpenAI

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For nearly 80 years, mathematicians have studied a deceptively simple question: if you place 𝑛 n points in the plane, how many pairs of points can be exactly distance 1 1 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 h

May 20, 2026 Research Milestone An OpenAI model has disproved a central conjecture in discrete geometry Read the proof (opens in a new window) Read the companion remarks (opens in a new window) Loading… Share For nearly 80 years, mathematicians have studied a deceptively simple question: if you place n n n points in the plane, how many pairs of points can be exactly distance 1 1 1 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 Re

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