A linear bound on some size Ramsey numbers for particular graphs

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:


For a graph \(G\), where \(G\) is \(Q_3\), \(K_{3,3}\) or \(H_5\) (formed by adding two vertex-disjoint chords to \(C_5\)), 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.