Chung–Graham conjecture for sparse halves in $K_4$-free graphs

Let $G$ be a $K_4$-free graph on $n$ vertices. Chung–Graham conjecture. $G$ contains a vertex set of size

$$\left\lfloor n/2 \right\rfloor$$

that spans at most $n^2/18$ edges. The bound is best possible, as shown by the Turán graph $T_3(n)$. The conjecture is proved in the paper for regular and almost regular graphs, but remains open for general $K_4$-free graphs.