Erdős Problem #1182 — Size-Ramsey extremal functions for triangles versus sparse graphs

Let $\hat r(G_1,G_2)$ be the least number of edges in a graph $H$ such that every red-blue colouring of $E(H)$ contains a red copy of $G_1$ or a blue copy of $G_2$. Let $f(n)$ be the largest $m$ for which some $n$-vertex graph $G$ with $m$ edges satisfies $\hat r(K_3,G)≤2n-1$, and let $F(n)$ be the largest $m$ such that every $n$-vertex graph with at most $m$ edges satisfies this inequality. Determine the orders of growth of $f(n)$ and $F(n)$. In particular, do $f(n)/F(n)→∞$ and $F(n)/n→∞$?