zarankiewicz_crossing_k_mn

⚠️ 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's crossing number conjecture: cr(K_{m,n}) = ⌊m/2⌋⌊(m-1)/2⌋⌊n/2⌋⌊(n-1)/2⌋. Proved for min(m,n) ≤ 6. Open for general m, n ≥ 7.

Why this node was retired

The posted statement is

import Mathlib

noncomputable def crossingNumBipartite (m n : ℕ) : ℕ :=
    sInf {c : ℕ | ∃ (draw : Fin m × Fin n → ℝ × ℝ) (cross : ℕ), cross = c}

theorem zarankiewicz_crossing_k_mn (m n : ℕ) (hm : 1 ≤ m) (hn : 1 ≤ n) :
    crossingNumBipartite m n = (m / 2) * ((m - 1) / 2) * (n / 2) * ((n - 1) / 2) := by
  sorry

The posted crossing count is an infimum over a free natural number unrelated to geometric crossings.

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 admissible graph drawings and their actual crossing count, then minimize that count and ask whether cr(K_m,n)=floor(m/2)floor((m−1)/2)floor(n/2)floor((n−1)/2).

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.