biclique_cover_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.

Graham–Pollak theorem: The minimum number of complete bipartite subgraphs covering all edges of K_n is n-1. For general graphs on n vertices, biclique cover number ≤ ⌈log₂n⌉+1. Exact tight bounds for specific graphs and variants are active research.

Why this node was retired

The posted statement is

import Mathlib

theorem biclique_cover_conjecture (n : ℕ) (hn : 2 ≤ n)
    (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] :
    ∃ (covers : Finset (Finset (Fin n) × Finset (Fin n))),
      (∀ e ∈ G.edgeFinset, ∃ bc ∈ covers, bc.1.Nonempty ∧ bc.2.Nonempty ∧
        ∀ a ∈ bc.1, ∀ b ∈ bc.2, a ∈ e ∨ b ∈ e) ∧
      covers.card ≤ Nat.clog 2 n + 1 := by
  sorry

The cross-pair predicate is not edge membership in a complete bipartite subgraph; the prose also confuses covering and partitioning.

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

Proposed corrected statement

For Graham–Pollak, define complete bipartite edge sets and require a partition of E(K_n), then prove the minimum number is n−1. If asking mere cover number instead, state that distinct problem with correct constants.

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.