Ramsey number of a cycle versus a graph of a given size

In this paper, we prove that for every $k$ and every graph $H$ with $m$ edges and no isolated vertices, the Ramsey number $R(C_k,H)$ is at most $2m+\lfloor \frac{k-1}{2}\rfloor$, provided $m$ is sufficiently large with respect to $k$. This settles a problem of Erdős, Faudree, Rousseau and Schelp.