Cycle–clique Ramsey numbers

We prove that $R(C_m,K_n)=(m-1)(n-1)+1$ for every pair of integers $m\ge n\ge 3$ 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.