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import FormalConjecturesUtil
open Filter Function Realopen scoped Pointwisenamespace Green53Suppose that $\mathbb{F}_2^n$ is partitioned in to sets $A_1, ..., A_K$. Does $2A_i$ contain a coset of codimension $O_K(1)$ for some $i$?
@[category research open, AMS 5 11]
theorem green_53 :
answer(sorry) ↔ ∃ (c : ℕ → ℕ), ∀ (n K : ℕ) (A : Fin K → Set (𝔽₂ n)),
(⋃ i, A i) = Set.univ →
Pairwise (Disjoint on A) →
∃ (i : Fin K) (S : AffineSubspace (ZMod 2) (𝔽₂ n)),
(S : Set (𝔽₂ n)) ⊆ A i + A i ∧
n ≤ Module.finrank (ZMod 2) S.direction + c K := ⊢ True ↔
∃ c,
∀ (n K : ℕ) (A : Fin K → Set (𝔽₂ n)),
⋃ i, A i = Set.univ →
Pairwise (Disjoint on A) → ∃ i S, ↑S ⊆ A i + A i ∧ n ≤ Module.finrank (ZMod 2) ↥S.direction + c K
All goals completed! 🐙-- TODO(jgd): Implement variants from Green's comments [Gr24].
end Green53