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?

1False ↔2  ∀ (δ : ℝ),3    0 < δ →4      ∃ c,5        0 < c ∧6          ∀ᶠ (n : ℕ) in Filter.atTop,7            ∀ (G : SimpleGraph (Fin n)),8              G.CliqueFree 5 → δ * ↑n ^ 2 ≤ ↑G.edgeFinset.card → ∃ S, c * ↑n ≤ ↑S.card ∧ G.CliqueFreeOn (↑S) 3