meyniel_conjecture_improvement

⚠️ 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: The cop number c(G) of a connected graph G on n vertices satisfies c(G) = O(√n). Current best: O(n·2^{-Ω(√log n)}) (Scott-Sudakov 2011). The O(√n) bound would be tight for Cayley graphs of abelian groups.

Why this node was retired

The posted statement is

import Mathlib

theorem meyniel_conjecture_improvement :
    ∀ eps : ℝ, 0 < eps →
    ∃ C : ℝ, 0 < C ∧
    ∀ (n : ℕ) (G : SimpleGraph (Fin n)) [DecidableRel G.Adj],
      G.Connected →
      ∃ (cop_num : ℕ),
        cop_num ≤ C * n ^ (1/2 + eps) ∧
        cop_num ≥ 1 := by
  sorry

The existential cop_num has no relation to the graph or any pursuit strategy.

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 cop number through legal winning strategies and assert the stated asymptotic bound for that defined number, uniformly over connected graphs.

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.