Cycle–clique Ramsey numbers

What OpenAI claims

Proves the Erdős–Faudree–Rousseau–Schelp conjecture: R(C_m,K_n)=(m-1)(n-1)+1 for every m ≥ n ≥ 3, except R(C₃,K₃)=6. This is the exact threshold forcing a red m-cycle or a blue n-clique in every red–blue coloring of a complete graph. (Lean)

Paper

1. Cycle--clique Ramsey numbers

   25 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract We prove that R(C_m,K_n)=(m-1)(n-1)+1 for every pair of integers m ≥ n ≥ 3 other than (m,n)=(3,3), for which R(C₃,K₃)=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.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.