Let $\delta > 0$. If $n$ is sufficiently large and $G$ is a graph on $n$ vertices with no $K_5$ and at least $\delta n^2$ edges, must $G$ contain a set of $\gg_\delta n$ vertices spanning no triangle?

The matching lower bound $\delta_3(5) \geq 1/12$, from the construction of Liu, Reiher, Sharifzadeh, and Staden [LRSS21] (improving the earlier $\delta_3(5) > 0$ of Balogh and Lenz [BaLe13]): for every $\epsilon, \delta > 0$ and all sufficiently large $n$ there is a $K_5$-free graph $G$ on $n$ vertices in which every triangle-free vertex set has at most $\delta n$ vertices, yet which has at least $(1/12 - \epsilon)n^2$ edges. In particular this refutes erdos_533.