Minimum Turán number over $k$-vertex, $l$-edge graphs: estimate $f(n;k,l)$ and its monotonicity (Erdős #766)

For integers $k$ and $l$, define $f(n;k,l)=\min_G \mathrm{ex}(n;G)$, where $G$ ranges over all graphs with exactly $k$ vertices and $l$ edges and $\mathrm{ex}(n;G)$ is the maximum number of edges in an $n$-vertex graph with no copy of $G$. Give good estimates for $f(n;k,l)$ in the range $k<l\le k^2/4$. In addition: for fixed $k$ and large $n$, is $f(n;k,l)$ a strictly monotone (increasing) function of $l$?

Acceptance. FULLY RESOLVES: (a) determine the asymptotic order of $f(n;k,l)$ throughout $k<l\le k^2/4$ (matching upper and lower bounds up to constants, with proof); AND (b) settle the monotonicity question — prove that for every fixed $k$ and all sufficiently large $n$, $f(n;k,l)$ is strictly increasing in $l$ on this range, or exhibit $k$ and $l$ with a proof that $f(n;k,l)=f(n;k,l+1)$ for arbitrarily large $n$. ADVANCES: rigorous new bounds on $f(n;k,l)$ for an infinite sub-range, improving on the trivial and Dirac–Erdős bounds in the background, with proof; or a verified computation of $f(n;k,l)$ across a range of small $k$ and $n$, with an exhaustiveness certificate over all $k$-vertex $l$-edge graphs, establishing or refuting strict monotonicity for those $k$. Deliver the proof, or the enumeration code plus the verified $f(n;k,l)$ table and monotonicity verdict.