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import FormalConjecturesUtilThe $\frac 1 3$–$\frac 2 3$ conjecture
namespace Conjecture_1_3_to_2_3
Does every finite partially ordered set that is not totally ordered contain two elements $x$ and $y$ such that the probability that $x$ appears before $y$ in a random linear extension is between $\frac 1 3$ and $\frac 2 3$?
The set of all total order extensions is represented as order preserving bijections $P$ of $1, ..., n$.
@[category research open, AMS 6]
theorem conjecture_1_3_to_2_3 : answer(sorry) ↔ ∀ (P : Type) [Finite P] [PartialOrder P]
(not_total : ¬ Std.Total (α := P) (· ≤ ·)) (total_ext : Set <| OrderHom P ℕ)
(total_ext_def : ∀ σ, σ ∈ total_ext ↔ Set.range σ = Set.Icc 1 (Nat.card P)),
∃ x y : P, ({σ ∈ total_ext | σ x < σ y}.ncard / total_ext.ncard : ℚ)
∈ Set.Icc (1/3) (2/3) := ⊢ True ↔
∀ (P : Type) [Finite P] [inst : PartialOrder P],
(¬Std.Total fun x1 x2 => x1 ≤ x2) →
∀ (total_ext : Set (P →o ℕ)),
(∀ (σ : P →o ℕ), σ ∈ total_ext ↔ Set.range ⇑σ = Set.Icc 1 (Nat.card P)) →
∃ x y, ↑{σ | σ ∈ total_ext ∧ σ x < σ y}.ncard / ↑total_ext.ncard ∈ Set.Icc (1 / 3) (2 / 3)
All goals completed! 🐙
end Conjecture_1_3_to_2_3