Let $\delta > 0$. If $n$ is sufficiently large and $G$ is a graph on $n$ vertices with no $K_{2,2,2}$ (the octahedron) and at least $\delta n^2$ edges, must $G$ contain an independent set of size $\gg_\delta n$?

1True ↔2  ∀ (δ : ℝ),3    0 < δ →4      ∃ c,5        0 < c ∧6          ∀ᶠ (n : ℕ) in Filter.atTop,7            ∀ (G : SimpleGraph (Fin n)),8              Erdos579.octahedron.Free G → δ * ↑n ^ 2 ≤ ↑G.edgeFinset.card → c * ↑n ≤ ↑G.indepNum