  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, Erd\H{o}s,
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 Erd\H{o}s, 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 Erd\H{o}s, Faudree,
Rousseau and Schelp.
