Let $n$ be some parameter tending to infinity. I am wondering does there exists some kind of graphs $G$ on vertex-set $[n]$ with maximum degree less than $D$, so that
- $D\ge n/w_1(n)$,
- $e_G$, the number of edges of $G$ satisfies $e_G\ge\frac{nD}{2w_2(n)}$,
- maximum codegree $\max_{u,v\in V_G:u\neq v}|\{w:wv,wu\in E_G\}|\le D/w_3(n)$,
where $w_1,w_2,w_3\to\infty$ as $n\to \infty$, and $w_1,w_2\ll w_3$ ? For $\ll$, I mean it will be good if $w_1,w_2$ are less than some power of $\log w_3$ or some small power of $w_3$. Definitely we assume $w_3$ less than some small power of $n$.