OpenAI model disproved a central conjecture in discrete geometry
Thank you for subscribing to BooksTopology and sticking with me. I have been writing about the interplay between artificial intelligence and pure mathematics, and how it can captivate funders and researchers. We’ve journeyed through the ideas of brilliant minds shaping this world - Fields Medalist Terence Tao, using AlphaEvolve to push boundaries, Fields Medalist Dr. Yau, Nobel Laureate Chen-Ning Yang, creator of Yang-Mills theory, Nobel Laureate Gerard ’t Hooft, Hong Wang and Joshua Zahl, who solved the Kakeya conjecture and won the Salem Prize, Fields Medalist Maryna Viazovska’s research paper with Apple LLM Evolutionary Search, Abel Prize winner Luis Caffarelli, AI Bourbaki and Modern Math (a mathematician group founded in 1934 by André Weil, Henri Cartan, Jean Dieudonné), Abel Prize Winner - Masaki Kashiwara’s new paper - Faithful action of braid group on bosonic extensions, etc.
For nearly 80 years, mathematicians have studied a deceptively simple question: if you place N points in the plane, how many pairs of points can be exactly distance 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.
Yesterday, OpenAI shared 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.
This breakthrough fundamentally rewrites the structural landscape of combinatorial geometry by shattering a lattice-based paradigm that went unchallenged for eighty years. By providing a strict polynomial improvement over Erdős’s original lower bound, the paper proves that traditional grid configurations are sub-optimal for maximizing unit distances. This refutation forces a deep re-evaluation of the deep-seated relationship between arithmetic structure and discrete geometric packings in the plane. Furthermore, it demonstrates that advanced computational models can uncover complex, non-obvious mathematical constructions that have eluded human intuition for generations. The explicit decoupling of this problem from standard integer lattices opens entirely new, uncharted pathways for attacking open problems in additive combinatorics. Ultimately, by shifting a canonical bounds threshold that stood for nearly a century, this work establishes a new baseline for modern extremal graph theory and discrete geometry.
Thanks,
Prasad.