  The Ramsey number $R(F,H)$ is the minimum number $N$ such that any $N$-vertex
graph either contains a copy of $F$ or its complement contains $H$. Burr in
1981 proved a pleasingly general result that for any graph $H$, provided $n$ is
sufficiently large, a natural lower bound construction gives the correct Ramsey
number involving cycles: $R(C_n,H)=(n-1)(\chi(H)-1)+\sigma(H)$, where
$\sigma(H)$ is the minimum possible size of a colour class in a
$\chi(H)$-colouring of $H$. Allen, Brightwell and Skokan conjectured that the
same should be true already when $n\geq |H|\chi(H)$.
  We improve this 40-year-old result of Burr by giving quantitative bounds of
the form $n\geq C|H|\log^4\chi(H)$, which is optimal up to the logarithmic
factor. In particular, this proves a strengthening of the
Allen-Brightwell-Skokan conjecture for all graphs $H$ with large chromatic
number.
