Does there exist some constant $c > 0$ such that for all sufficiently large $n$, if $G$ is a graph with $n$ vertices and at least $(1/8 - c)n^2$ edges then $G$ must contain either a $K_4$ or an independent set on at least $n/\log n$ vertices?

Does there exist some constant $c > 0$ such that for all sufficiently large $n$, if $G$ is a graph with $n$ vertices and at least $(1/8 - c)n^2$ edges then $G$ must contain either a $K_4$ or an independent set on at least $n/\log n$ vertices?

The answer is no, as shown by Fox, Loh, and Zhao [FLZ15].