Let $f(n)$ be maximal such that there is a connected graph $G$ with $n$ vertices and $f(n)$ edges such that$$R(K_3,G)=2n-1.$$Let $F(n)$ be maximal such that every connected graph $G$ with $n$ vertices and $\le F(n)$ edges has$$R(K_3,G)=2n-1.$$Estimate $f(n)$ and $F(n)$. In particular, is it true that $F(n)/n\to \infty$?