Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 165-02. The crossing number of complete bipartite graphs
Manuscript map · lines 5495–5504
OpenAI@2026-10-07#rem:oai-p-165-02
Authors: OpenAI. Family 165.
Title: The crossing number of complete bipartite graphs.
Abstract: 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 ${}displaystyle {}left{}lfloor{}frac m2{}right{}rfloor {}left{}lfloor{}frac{m-1}{2}{}right{}rfloor {}left{}lfloor{}frac n2{}right{}rfloor {}left{}lfloor{}frac{n-1}{2}{}right{}rfloor.$
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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