A linear bound on some size Ramsey numbers

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 $$\hat{r}(G)$$.

The following problem is due to Erdös, Faudree, Rousseau and Schelp [2], [1].

A Ramsey size linear problem (proposed by Erdös, Faudree, Rousseau and Schelp [1])

Suppose a graph $$G$$ satisfies the property that every subgraph of $$G$$ on $$p$$ vertices has at most $$2p-3$$ edges. Is it true that, for any graph $$H$$ on $$n$$ edges, $$r(G,H) \leq cn?$$

In [1], it was shown that for a graph $$G$$ with $$p$$ vertices and $$q$$ edges, we have

$$\displaystyle r(G,K_n) > c (n / \log n)^{(q-1)/(p-2)}$$

for $$n$$ sufficiently large.

This implies that for a graph $$G$$ with $$p$$ vertices and $$2p-2$$ edges, the inequality (1) does not hold for all $$H$$ with $$n$$ edges.

In the other direction, in [1] it was shown that for any graph $$G$$ with $$p$$ vertices and at most $$p+1$$ edges, (1) holds.