theorem
Green12.green_12 :
sorry ↔ ∀ {G : Type u_1} [inst : AddCommGroup G] [inst_1 : Fintype G] [inst_2 : DecidableEq G] (A : Finset G),
have N := Fintype.card G;
have α := ↑A.card / ↑N;
have valid_tuples := {t : (Fin 5 → G) × (Fin 5 → G) | ∀ (i j : Fin 5), j ∈ {i, i + 1, i + 2} → t.1 i + t.2 j ∈ A};
↑valid_tuples.card ≥ α ^ 15 * ↑N ^ 10
Let $G$ be an abelian group of size $N$, and suppose that $A \subset G$ has density $\alpha$. Are there at least $\alpha^{15} N^{10}$ tuples $(x_1, \dots, x_5, y_1, \dots, y_5) \in G^{10}$ such that $x_i + y_j \in A$ whenever $j \in \{i, i+1, i+2\}$?
Note: We interpret indices modulo 5.