sidorenko_conjecture

⚠️ 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.

Sidorenko's conjecture (1991): For every bipartite graph H and graph G on n vertices with p edges per vertex pair, the number of labeled copies of H in G is at least n^{v(H)} · p^{e(H)}. Proved for trees, even cycles, complete bipartite graphs, and many other cases. General bipartite case open.

Why this node was retired

The posted statement is

import Mathlib

theorem sidorenko_conjecture (m n : ℕ) (hm : 1 ≤ m) (hn : 1 ≤ n)
    (H : SimpleGraph (Fin m ⊕ Fin n)) [DecidableRel H.Adj]
    (hbip : ∀ v : Fin m ⊕ Fin n, ∀ w : Fin m ⊕ Fin n,
      H.Adj v w → (v.isLeft ↔ ¬w.isLeft))
    (p : ℝ) (hp : 0 ≤ p) (hp1 : p ≤ 1) :
    ∀ (k : ℕ) (G : SimpleGraph (Fin k)) [DecidableRel G.Adj],
      (G.edgeFinset.card : ℝ) / (k * (k - 1) / 2) = p →
      (∃ count : ℕ, count = {f : (Fin m ⊕ Fin n) → Fin k | ∀ u v, H.Adj u v → G.Adj (f u) (f v)}.ncard) →
      p ^ H.edgeFinset.card ≤
        {f : (Fin m ⊕ Fin n) → Fin k | ∀ u v, H.Adj u v → G.Adj (f u) (f v)}.ncard /
        (k : ℝ) ^ (m + n) := by
  sorry

The left side uses distinct-pair edge density while the right side uses vertex-map homomorphism density.

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

Proposed corrected statement

Define p=2*|E(G)|/k² for k>0, consistent with t(K2,G); keep the homomorphism-density denominator k^|V(H)|.

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.