Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 189-01. Cycle–clique Ramsey numbers
Manuscript map · lines 5825–5834
OpenAI@2026-10-07#rem:oai-p-189-01
Authors: OpenAI. Family 189.
Title: Cycle–clique Ramsey numbers.
Abstract: We prove that $R(C_m,K_n)=(m-1)(n-1)+1$ for every pair of integers $m{}ge n{}ge3$ other than $(m,n)=(3,3)$, for which $R(C_3,K_3)=6$. This establishes the cycle–clique conjecture of Erdős, Faudree, Rousseau and Schelp. The proof combines expansion in a minimal counterexample with a large-clique lemma and an optimization of paths joining clique vertices. These arguments reduce the remaining cases to $3{,}099$ finite parameter-pattern instances, which are excluded by two exact implementations of proved inference rules. Complete programs and deduction traces accompany the paper.
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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