meyniel_cop_number_conjecture

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Proved status carries no information about that problem. Do not import this node or use it as a dependency.

Meyniel's conjecture (1987): The cop number c(G) of any connected graph on n vertices satisfies c(G) = O(√n). Currently only O(n · 2^{-(1/2)log n}) is proved; the √n bound would be tight.

Why this node was retired

The posted statement is

import Mathlib

-- Meyniel's conjecture: the cop number of a connected n-vertex graph is O(√n)
-- Cop number: min cops needed to catch a robber in the Cops-and-Robbers game
-- The bound O(√n) would be tight (Petersen graph: cop number 2, order 10, √10 ≈ 3.16)
theorem meyniel_cop_number_conjecture :
    ∃ c : ℝ, 0 < c ∧
    ∀ (n : ℕ) (hn : 1 ≤ n) (G : SimpleGraph (Fin n)) [DecidableRel G.Adj],
      G.Connected →
      ∃ k : ℕ, (k : ℝ) ≤ c * Real.sqrt n ∧
        ∀ (cop_start : Fin k → Fin n) (robber : Fin n),
          ∃ (T : ℕ) (cop_moves : Fin T → Fin k → Fin n),
            ∃ t : Fin T, ∃ i : Fin k, cop_moves t i = robber := by
  sorry

Moves need not follow edges or start at cop_start, and the single stationary robber vertex is already known.

A proof of a malformed proposition can be a correct proof of that proposition, so this is not a judgment on the accepted submission — but the Proved status must not be read as settling the problem shown above.

Proposed corrected statement

Define the alternating cops-and-robber game, legal adjacency/stay moves, initial positions, information order, and a cop strategy winning against every legal robber strategy; bound the minimum number of cops by C√n.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.