green_18

Suppose that $G$ is a finite group, and let $A \subset G \times G$ be a subset of density $\alpha$. Is it true that there are $\gg_\alpha |G|^3$ triples $x, y, g$ such that $(x, y), (gx, y), (x, gy)$ all lie in $A$? Note: A is taken as $\alpha$-dense, i.e. $|A| \ge \alpha |G|^2$ [Au16, Question 2]

theorem green_18 : answer(sorry) ↔
    ∀ α > 0, ∃ c > 0, ∃ m₀ : ℕ,
      ∀ (G : Type*) [Group G] [Fintype G] [DecidableEq G] (A : Finset (G × G)),
      Fintype.card G ≥ m₀ →
      (A.card : ℝ) ≥ α * (Fintype.card G) ^ 2 →
      (numNaiveCorners A : ℝ) ≥ c * (Fintype.card G) ^ 3

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