A linear bound on some size Ramsey numbers for trees

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:


Suppose \(r(G, T_n) \leq cn\) for any tree \(T_n\) on \(n\) vertices and \(r(G,K_n) \leq cn^2\). Is it true that \[ r(G,H) \leq c n \] for any graph \(H\) with \(n\) edges?

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