expander_mixing_lemma_tight

⚠️ 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.

Expander mixing lemma: For k-regular graphs with second eigenvalue ≤ λ, edges between sets S and T deviate from expected by ≤ λ√(|S||T|). Constructing explicit Ramanujan graphs (λ = 2√(k-1)) for all k is an open problem.

Why this node was retired

The posted statement is

import Mathlib

theorem expander_mixing_lemma_tight (n k : ℕ) (hn : 3 ≤ n) (hk : 2 ≤ k)
    (G : SimpleGraph (Fin n)) [DecidableRel G.Adj]
    (hreg : ∀ v : Fin n, G.degree v = k)
    (lambda2 : ℝ) (hlambda : 0 ≤ lambda2 ∧ lambda2 < k) :
    ∀ (S T : Finset (Fin n)),
      |(G.edgeFinset.filter (fun e => ∃ a ∈ S, ∃ b ∈ T, a ∈ e ∧ b ∈ e)).card -
        (k : ℝ) * S.card * T.card / n| ≤
        lambda2 * Real.sqrt ((S.card : ℝ) * T.card) := by
  sorry

The lambda parameter is unrelated to the spectrum, and the edge filter miscounts edges when S and T overlap.

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

Proposed corrected statement

Assume an operator-norm/eigenvalue bound for the adjacency operator on constants-perpendicular vectors, and count ordered adjacent pairs in S×T in the discrepancy.

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.