Source labels openChecked July 26, 2026

Union-closed sets conjecture: Cardinality Even Of Union Closed Tight

If the UC conjecture is tight for some family A then $|A| = 2^k$ for some $k$ .

Combinatorics AMS 5

Mathematical statement

If the UC conjecture is tight for some family A then $|A| = 2^k$ for some $k$ .

Reference: Conjecture 3 in https://www.nieuwarchief.nl/serie5/pdf/naw5-2023-24-4-225.pdf.

Statement source: Wikipedia statement material

Statement terms: CC-BY-SA-4.0

Attributed source material. Reuse must follow the linked attribution and share-alike terms.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

union_closed.variants.cardinality_even_of_union_closed_tight

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem union_closed.variants.cardinality_even_of_union_closed_tight    [Nonempty n] (hA : A ≠ {∅} ∧ A ≠ ∅) (hA : IsUnionClosed A)    (UCC_tight : ∀ i, #{x ∈ A | i ∈ x} = (1 / 2 : ℝ) * #A) :    ∃ k, #A = 2 ^ k := by  sorry

Statement source: Formal Conjectures
Lean version: v4.27.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Pinned Formal Conjectures sourceformalization Wikipediabackground