bollobas_set_pairs

⚠️ 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.

Bollobás set-pairs inequality (1965): If (Aᵢ, Bᵢ) are cross-intersecting disjoint pairs, then ∑ C(|Aᵢ|+|Bᵢ|, |Aᵢ|)⁻¹ ≤ 1. Fundamental result in extremal set theory.

Why this node was retired

The posted statement is

import Mathlib

theorem bollobas_set_pairs {α : Type*} [Fintype α] [DecidableEq α]
    (pairs : Finset (Finset α × Finset α))
    (hdisjoint : ∀ p ∈ pairs, Disjoint p.1 p.2)
    (hcross : ∀ p ∈ pairs, ∀ q ∈ pairs, p ≠ q →
      (p.1 ∩ q.2).Nonempty ∨ (q.1 ∩ p.2).Nonempty) :
    pairs.card ≤ Nat.choose (Fintype.card α) (Fintype.card α / 2) := by
  sorry

The cross-intersection hypothesis uses an OR, permitting one direction to be empty.

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 A_i∩B_j to be nonempty for every distinct ordered pair i,j, while each A_i∩B_i is empty; state the intended weighted Bollobás inequality or its derived cardinality consequence.

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.