zarankiewicz_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.

Zarankiewicz problem / Kővári–Sós–Turán conjecture: The maximum number of edges in a K_{s,t}-free bipartite graph on (n,n) vertices satisfies z(n,n;s,t) ≤ O(n^{2-1/s}). For s=t=2, the conjectured exact formula gives ≤ (1/2)(n^{3/2} + n).

Why this node was retired

The posted statement is

import Mathlib

-- Zarankiewicz problem: z(n,n;s,t) = ex(n,n; K_{s,t})
-- Conjecture: z(n,n;2,2) = (1/2)(1 + sqrt(4n-3)) * n ~ (1/2) * n^{3/2}
-- Formulated as: every bipartite K_{2,2}-free graph on n+n vertices has ≤ (1/2)(n^{3/2} + n) edges
theorem zarankiewicz_conjecture (n : ℕ) (hn : 1 ≤ n) :
    ∀ (G : SimpleGraph (Fin n ⊕ Fin n)) [DecidableRel G.Adj],
      (∀ s₁ s₂ : Fin n, ∀ t₁ t₂ : Fin n, s₁ ≠ s₂ → t₁ ≠ t₂ →
        ¬(G.Adj (Sum.inl s₁) (Sum.inr t₁) ∧ G.Adj (Sum.inl s₁) (Sum.inr t₂) ∧
          G.Adj (Sum.inl s₂) (Sum.inr t₁) ∧ G.Adj (Sum.inl s₂) (Sum.inr t₂))) →
      2 * G.edgeFinset.card ≤ n ^ 2 + n * Nat.sqrt n := by
  sorry

The graph on two summands is never required to be bipartite; the stated numerical formula also differs from the source asymptotic 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

Require every edge to cross the two parts and use a precise valid K2,2 extremal inequality, e.g. |E|≤n(1+sqrt(4n−3))/2 over ℝ, rather than mixing n² and a floored square root.

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.