The Erdős-Faudree-Rousseau-Schelp cycle-clique conjecture: $R(C_m,K_n)=(m-1)(n-1)+1$ for $m\ge n\ge3$

Theorem 1: $R(C_m,K_n)=(m-1)(n-1)+1$ for every $m\ge n\ge3$ other than $(3,3)$, where $R(C_3,K_3)=6$. The proof takes a minimal counterexample (independent sets expand), finds a clique of order at least $\max\{3,\lfloor k/2\rfloor\}$, optimizes path systems between clique vertices, and reduces the rest to 3,099 finite instances excluded by two exact programs. Keevash-Long-Skokan had already left only finitely many pairs open, without an explicit list; this result closes all of them. It does not address the side questions recorded with Problem #551 for fixed $n$ (the least cycle length at which the identity holds when $m<n$ is allowed, and the minimum of $R(C_k,K_n)$ over $k$).

Theorem 1: $R(C_m,K_n)=(m-1)(n-1)+1$ for every $m\ge n\ge3$ other than $(3,3)$, where $R(C_3,K_3)=6$. The proof takes a minimal counterexample (independent sets expand), finds a clique of order at least $\max\{3,\lfloor k/2\rfloor\}$, optimizes path systems between clique vertices, and reduces the rest to 3,099 finite instances excluded by two exact programs. Keevash-Long-Skokan had already left only finitely many pairs open, without an explicit list; this result closes all of them. It does not address the side questions recorded with Problem #551 for fixed $n$ (the least cycle length at which the identity holds when $m<n$ is allowed, and the minimum of $R(C_k,K_n)$ over $k$).