A linear bound on some size Ramsey numbers for odd cycles

The size Ramsey number \( \hat{r}(G,H)\) is the least integer \( m\) for which there exists a graph \( F\) with \( m\) edges so that in any coloring of the edges of \( F\) in red and blue, there is always either a red copy of \( G\) or a blue copy of \( H\). Sometimes we write \( F \rightarrow (G,H)\) to denote this. For \( G=H\), we denote \( \hat{r}(G,G)\) by \( \hat{r}(G)\).

In [1], Erdös, Faudree, Rousseau and Schelp raised the following problem:


What is the best constant \(c\) satisfying \[ r(C_{2k+1}, H) \leq c(2k+1)n \] where \(H\) is any graph on \(n\) edges without isolated vertices?

1 P. Erdös, R. Faudree, C. C. Rousseau and R. H. Schelp, Ramsey size linear graphs, Combin. Probab. Comput. 2 (1993), 389-399.