Cycle-Complete Ramsey Numbers

Abstract

Abstract The Ramsey number $r(C_{\ell },K_n)$ is the smallest natural number $N$ such that every red/blue edge colouring of a clique of order $N$ contains a red cycle of length $\ell $ or a blue clique of order $n$. In 1978, Erd̋s, Faudree, Rousseau, and Schelp conjectured that $r(C_{\ell },K_n) = (\ell -1)(n-1)+1$ for $\ell \geq n\geq 3$ provided $(\ell ,n) \neq (3,3)$. We prove that, for some absolute constant $C\ge 1$, we have $r(C_{\ell },K_n) = (\ell -1)(n-1)+1$ provided $\ell \geq C\frac{\log n}{\log \log n}$. Up to the value of $C$ this is tight since we also show that, for any $\varepsilon>0$ and $n> n_0(\varepsilon )$, we have $r(C_{\ell }, K_n) \gg (\ell -1)(n-1)+1$ for all $3 \leq \ell \leq (1-\varepsilon )\frac{\log n}{\log \log n}$. This proves the conjecture of Erd̋s, Faudree, Rousseau, and Schelp for large $\ell $, a stronger form of the conjecture due to Nikiforov, and answers (up to multiplicative constants) two further questions of Erd̋s, Faudree, Rousseau, and Schelp.