OpenAI research collection · 2 manuscripts · Source checked 2026-10-19
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.
Short excerpts below come from OpenAI's abstracts. The full abstract and original PDF preserve the paper's precise scope.
We prove the Harary–Hill conjecture: for every positive integer n, the ordinary crossing number of the complete graph K_(n) is \displaystyle \frac14\lfloor\frac n2\rfloor \lfloor\frac{n-1}{2}\rfloor \lfloor\frac{n-2}{2}\rfloor \lfloor\frac{n-3}{2}\rfloor.
This PDF is listed in OpenAI's formalization catalogue. See the scope and dependencies before interpreting coverage.
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 \lfloor\frac m2\rfloor \lfloor\frac{m-1}{2}\rfloor \lfloor\frac n2\rfloor \lfloor\frac{n-1}{2}\rfloor.
This PDF is listed in OpenAI's formalization catalogue. See the scope and dependencies before interpreting coverage.
New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.
Explore the AGI Clock ↗