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Green's open problem 12 Let G G G be an abelian group of size N N N , and suppose that A ⊂ G A \subset G A ⊂ G has density α \alpha α .
Are there at least α 15 N 10 \alpha^{15} N^{10} α 15 N 10 tuples
( x 1 , … , x 5 , y 1 , … , y 5 ) ∈ G 10 (x_1, \dots, x_5, y_1, \dots, y_5) \in G^{10} ( x 1 , … , x 5 , y 1 , … , y 5 ) ∈ G 10
such that
x i + y j ∈ A x_i + y_j \in A x i + y j ∈ A whenever
j ∈ { i , i + 1 , i + 2 } j \in \{i, i+1, i+2\} j ∈ { i , i + 1 , i + 2 } ?
Note: We interpret indices modulo 5.
Two ways to claim this Each is a separate task with its own bundle and its own bounty. Pick the one your proof argues for.
Bounty
$4,761
paid on an accepted proof
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Lean type
True ↔
∀ {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 | ∀ (i j : Fin 5), j ∈ {i, i + 1, i + 2} → t.1 i + t.2 j ∈ A};
↑valid_tuples.card ≥ α ^ 15 * ↑N ^ 10What you must prove
import FormalConjectures.GreensOpenProblems.«12»
import TaskSupport
namespace Bounty
theorem target : ¬ (fcTypeOfName% "Green12.green_12") := by
sorry
end Bounty
Pinned source: FormalConjectures/GreensOpenProblems/12.lean
Source type SHA-256 sha256:20e4a5d0e1c32e743821fd78401f7b8743e3049569a071ef6abb3342cfc9891b
Task id fc-379fc029-green12-green-12-f5a5026dae-counterexample-v1
Task commitment sha256:925981cabb1e110ef15a45b8c636abd3b597e67d397a5080bf678ce9b9cb89f9 Something wrong with this formalization?
A statement that does not faithfully capture the original conjecture is the one real risk here, so we would rather hear about it early - before someone spends weeks on it.
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Green's open problem 12 · Conjectures.io