{"id":"openai-math/Cycle-clique-Ramsey-numbers","pdf":"https://github.com/openai/math/blob/main/preprints/Cycle-clique-Ramsey-numbers-September-25-2026/Cycle-clique-Ramsey-numbers-September-25-2026.pdf","url":"https://github.com/openai/math/tree/main/preprints/Cycle-clique-Ramsey-numbers-September-25-2026","tags":["ramsey-theory","extremal-combinatorics","graph-theory"],"score":0,"title":"Cycle--clique Ramsey numbers","domain":"Combinatorics","family":"189","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.CO","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-25","reactions":{},"formalized":false,"family_title":"Cycle–clique Ramsey numbers"}