OpenAI disproves the Erdős unit distance conjecture: an 80-year-old open problem ended by an AI counterexample

## OpenAI disproves the Erdős unit distance conjecture In 1946, Paul Erdős posed the unit distance problem: given n points in the plane, what is the maximum number of pairs of points whose distance is exactly 1? For nearly 80 years, mathematicians widely believed that the square grid configuration was optimal. OpenAI announced that its internal reasoning model has produced the first fully autonomous counterexample proof. **Key points** - **A counterexample, not a proof**: The model did not prove that the grid is optimal. Instead it constructed a counterexample: an infinite family of point sets that beat the grid bound by an explicit polynomial factor (refined to n^1.014 by Sawin of Princeton). - **Weighty assessments**: Tim Gowers, a 1998 Fields Medalist, called it "a milestone for AI mathematics," saying no previous AI-generated proof had come close to this level; Gil Kalai compared it to the 1976 computer-assisted proof of the four color theorem. - **A win for general-purpose models**: The key detail is that this is a general-purpose reasoning model, neither trained specifically for mathematics nor targeted at this problem. It found, on its own, a path that generations of combinatorial geometers had missed. - **Controversy**: Andreas Thom has accused OpenAI of using his unpublished research, and the dispute is still unfolding; when citing this work, it is advisable to present it alongside the result. > Source: [read the original](https://pasqualepillitteri.it/en/news/3065/openai-erdos-unit-distance-conjecture-ai-breakthrough). This text is a summary for reference only.