Andrea Freschi: Ramsey number of a cycle versus a graph of given size

Let $C_k$ be the cycle on $k$ vertices and $H$ be an arbitrary graph on $m$ edges and no isolated vertices. 

Erdős, Faudree, Rousseau and Schelp conjectured that if $m$ is sufficiently large compared to $k$ then the Ramsey number $R(C_k,H)$ is at most $2m+\left\lfloor\frac{k-1}{2}\right\rfloor$. 

This was verified for $k$ even and for $k\in\{3,5\}$.