OPEN

Let $k\geq 3$. Is it true that, for any graph $H$ on $m$ edges without isolated vertices,
\[R(C_k,H) \leq 2m+\left\lceil\frac{k-1}{2}\right\rceil?\]

This was proved for even $k$ by Erdős, Faudree, Rousseau, and Schelp

[EFRS93]. It was proved for $k=3$ by Sidorenko

[Si93].

See also the entry in the graphs problem collection.