  Given a pair of graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the
smallest $N$ such that every red-blue coloring of the edges of the complete
graph $K_N$ contains a red copy of $G$ or a blue copy of $H$. If a graph $G$ is
connected, it is well known and easy to show that $R(G,H) \geq
(|G|-1)(\chi(H)-1)+\sigma(H)$, where $\chi(H)$ is the chromatic number of $H$
and $\sigma(H)$ is the size of the smallest color class in a $\chi(H)$-coloring
of $H$. A graph $G$ is called $H$-good if $R(G,H)=
(|G|-1)(\chi(H)-1)+\sigma(H)$. The notion of Ramsey goodness was introduced by
Burr and Erd\H{o}s in 1983 and has been extensively studied since then.
  In this paper we show that if $n\geq 10^{60}|H|$ and $\sigma(H)\geq
\chi(H)^{22}$ then the $n$-vertex cycle $C_n$ is $H$-good. For graphs $H$ with
high $\chi(H)$ and $\sigma(H)$, this proves in a strong form a conjecture of
Allen, Brightwell, and Skokan.
