Discrete Math Reading Seminar

Speaker: Nick Layman
Title: Open AI Crossing number of Knn
Abstract: On October 6, 2026 OpenAI released a preprint of a paper claiming to have exactly determined the crossing number of all complete bipartite graphs. This problem, and the conjectured answer by Zerankiewicz, has been actively worked on since 1954 when Zerankiewicz first proposed a proof which was later found to contain an error. Small examples and families with one part small were solved, but global progress was restricted to incrementally improving asymptotic lower bounds. OpenAI’s paper develops new tools in linear algebra to solve the problem outright in its entirety. I will attempt to describe how those tools are applied to graph drawings and crossing numbers, but I will not be describing the proofs of those tools themselves.