cayley_graph_diameter

⚠️ Retired — specification defect

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

Babai–Seress diameter conjecture: Every Cayley graph of a finite group G has diameter O((log|G|)²). Proved by Helfgott–Seress (2014). Conjectured to be O(log|G|) for non-abelian simple groups.

Why this node was retired

The posted statement is

import Mathlib

theorem cayley_graph_diameter (G : Type*) [Group G] [Fintype G]
    (S : Finset G) (hS : ∀ s ∈ S, s⁻¹ ∈ S) (hgen : Subgroup.closure (S : Set G) = ⊤) :
    ∃ (diam : ℕ),
      (diam : ℝ) ≤ (Real.log (Fintype.card G))^2 ∧
      ∀ g : G, ∃ (word : List G),
        word.length ≤ diam ∧
        word.prod = g ∧
        ∀ s ∈ word, s ∈ S := by
  sorry

A coefficient-one log² bound is imposed for every finite group, without the group-class/asymptotic restrictions needed by the named diameter theme.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

For the standard Babai-type problem, restrict to nonabelian finite simple groups and ask for absolute constants C,c with diameter≤C(log|G|)^c for every symmetric generating set. Merely adding a constant does not fix the all-finite-groups statement, since cyclic groups give linear diameter.

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.