Let $\epsilon > 0$ and let $n$ be sufficiently large depending on $\epsilon$. Is there a graph on $n$ vertices with at least $n^2/8$ many edges which contains no $K_4$, such that the largest independent set has size at most $\epsilon n$?

The quantitative strengthening proved by Fox, Loh, and Zhao [FLZ15]: there is a constant $C > 0$ such that for all sufficiently large $n$ there exists a $K_4$-free graph on $n$ vertices with at least $n^2/8$ edges whose largest independent set has size at most $$C \cdot \frac{(\log\log n)^{3/2}}{(\log n)^{1/2}} \cdot n.$$