hedetniemi_conjecture_disproof

⚠️ 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.

Hedetniemi's conjecture disproof (Shitov 2019): There exist graphs G, H with χ(G) = χ(H) = 4 but χ(G × H) < 4, disproving the 1966 conjecture χ(G × H) = min(χ(G), χ(H)).

Why this node was retired

The posted statement is

import Mathlib

theorem hedetniemi_conjecture_disproof :
    ∃ (n m : ℕ) (G : SimpleGraph (Fin n)) (H : SimpleGraph (Fin m))
        (hDG : DecidableRel G.Adj) (hDH : DecidableRel H.Adj),
      @SimpleGraph.chromaticNumber _ G = 4 ∧
      @SimpleGraph.chromaticNumber _ H = 4 ∧
      ∃ (hDP : DecidableRel (G.boxProd H).Adj),
        @SimpleGraph.chromaticNumber _ (G.boxProd H) < 4 := by
  sorry

The target uses Cartesian boxProd instead of the categorical product, and the fixed chromatic value4 does not express the general counterexample.

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

Proposed corrected statement

Use the categorical graph product (adjacency in both coordinates) and ask for finite graphs with χ(G×H)<min(χ(G),χ(H)), without the unsupported fixed4 requirement.

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.