Maximum edges keeping $R(K_3,G)=2n-1$: estimate $f(n)$ and $F(n)$ (Erdős #1182)

For a connected graph $G$ on $n$ vertices, the Ramsey number $R(K_3,G)$ is the least $N$ such that every red/blue colouring of the edges of $K_N$ contains a red triangle $K_3$ or a blue copy of $G$; one always has $R(K_3,G)\geq 2n-1$. Let $f(n)$ be the maximum number of edges of a connected graph $G$ on $n$ vertices with $R(K_3,G)=2n-1$, and let $F(n)$ be the maximum $M$ such that every connected $n$-vertex graph with at most $M$ edges satisfies $R(K_3,G)=2n-1$. Estimate $f(n)$ and $F(n)$; in particular, is it true that $F(n)/n\to\infty$?

Acceptance. FULLY RESOLVES: determine the asymptotics of both $f(n)$ and $F(n)$ — e.g. prove $F(n)=\Theta(n)$ with the correct constant (settling $F(n)/n\to\infty$, expected negatively by Brandt) and pin down the growth exponent of $f(n)$ between the stated $n^{3/2}(\log n)^{1/2}$ and $n^{5/3}(\log n)^{2/3}$ — with complete proofs (machine-checkable or full written). ADVANCES: (a) improve any of the stated bounds — the lower bound $F(n)\geq\frac{17n+1}{15}$, the upper bound $F(n)\leq 84n$, or either $f(n)$ bound — strictly beyond the background, with proof or a reproducible extremal construction; (b) compute new exact values of $f(n)$ and/or $F(n)$ for $n$ beyond the tabulated range ($F: 1,2,5,7,8,\ldots$; $f: 1,2,5,8,12,\ldots$) via exhaustive graph enumeration with an exhaustiveness certificate and the underlying $R(K_3,G)$ checks; or (c) resolve the analogous $K_m$ question for a new value of $m$. Deliver the proof or improved-bound construction, or the enumeration code plus certified new table values.