The – conjecture
The – conjecture
Problem statement
conjecture_1_3_to_2_3
Does every finite partially ordered set that is not totally ordered contain two elements and such that the probability that appears before in a random linear extension is between and ?
The set of all total order extensions is represented as order preserving bijections of .
Lean 4
Formal statement
Lean 4/-
Copyright 2025 The Formal Conjectures Authors.
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
https://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/
import FormalConjecturesUtil
/-!
# The $\frac 1 3$–$\frac 2 3$ conjecture
*Reference:* [Wikipedia](https://en.wikipedia.org/wiki/1/3%E2%80%932/3_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) := by
sorry
end Conjecture_1_3_to_2_3
On-site verification needs an executable Lean target. None is attached yet.
A source attachment or link is not a verified proof. The statement’s correspondence with the original question needs human review.
Proofs & verification
No Lean has been submitted for this problem yet. A Lean file can be attached to a submission for review, though with no target here it cannot be machine-checked.
Background
Source snapshot inspected on 10 September 2026. Subsequent research status and the accuracy of source formalizations have not been independently certified.
Fixed commit b82b08faa9006484021c12005ab41287fb2ffb69. This file contains 1 source-labeled research-open declarations. Other declarations in the attached file can be definitions, solved results or tests. The file is preserved as source material; it has not been compiled or proved by this import.
Copyright The Formal Conjectures Authors. Code: Apache-2.0; other original material: CC BY 4.0. Third-party sources retain their own licenses. Original source files and notices preserved.
Recorded progress
Add yours →Nothing recorded yet. A pointer to a paper the entry is missing counts.
Machine attempts
0 recordedAttempts run against the statement below and are recorded in full, including the failures. The point is to stop the next model — and the next person — from re-walking a road that is already known to end. Each attempt costs 25 credits.
No machine attempts recorded.
Sign in to run an attempt. New accounts start with enough credits for ten.
Discussion
0 notesNotes are for clarifying the statement, flagging a missing reference, or recording why an approach fails. Claimed solutions belong in a submission, where they can be reviewed.
No notes yet.
Sign in to add a note.