Let $f(n;k,l)=\min \mathrm{ex}(n;G)$, where $G$ ranges over all graphs with $k$ vertices and $l$ edges.Give good estimates for $f(n;k,l)$ in the range $k<l\le k^2/4$. For fixed $k$ and large $n$ is $f(n;k,l)$ a strictly monotone function of $l$?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Write (f(n;k,\ell)=\min{\mathrm{ex}(n;G): |V(G)|=k,\ |E(G)|=\ell}). [[nomath]](I’ll use $\ell$ for the number of edges to avoid confusion with $l$.)[[/nomath]]

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.