Resolves the Harary–Hill conjecture and Turán's brickyard problem in the Zarankiewicz formulation, determining the crossing numbers of every complete and complete bipartite graph. The result proves the optimality of the classical drawings among all plane drawings with continuous edge arcs. (Lean)
Manuscripts (2)
The crossing number of complete graphs
September 23, 2026 · source · Lean formalization available · cite as \cite{OAI:The-crossing-number-of-complete-graphs-September-23-2026}
We prove the Harary–Hill conjecture: for every positive integer n, the ordinary crossing number of the complete graph Kn is
\displaystyle \frac14\left\lfloor\frac n2\right\rfloor \left\lfloor\frac{n-1}{2}\right\rfloor \left\lfloor\frac{n-2}{2}\right\rfloor \left\lfloor\frac{n-3}{2}\right\rfloor.
The crossing number of complete bipartite graphs
September 23, 2026 · source · Lean formalization available · cite as \cite{OAI:The-crossing-number-of-complete-bipartite-graphs-September-23-2026}
We prove the Zarankiewicz crossing-number conjecture, resolving Turán's brickyard problem. For all positive integers m, n, the ordinary crossing number of the complete bipartite graph K_{m,n} is
\displaystyle \left\lfloor\frac m2\right\rfloor \left\lfloor\frac{m-1}{2}\right\rfloor \left\lfloor\frac n2\right\rfloor \left\lfloor\frac{n-1}{2}\right\rfloor.
Result 165 · Combinatorics. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md