frankl_union_closed_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.

Frankl's union-closed conjecture (1979): In any finite union-closed family of sets, there exists an element that belongs to at least half of the sets. Significant progress by Gilmer (2022) showed a constant fraction ≥ 0.01; the sharp 1/2 bound remains open.

Why this node was retired

The posted statement is

import Mathlib

theorem frankl_union_closed_conjecture {α : Type*} [DecidableEq α] [Fintype α]
    (F : Finset (Finset α)) (hF : F.Nonempty)
    (hclosed : ∀ A ∈ F, ∀ B ∈ F, A ∪ B ∈ F) :
    ∃ x : α, 2 * (F.filter (fun A => x ∈ A)).card ≥ F.card := by
  sorry

A nonempty family may consist only of the empty set.

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 that at least one member of F is nonempty (equivalently the union of F is nonempty); retain the half-frequency conclusion.

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.