All open problems
Open problemChecked July 24, 2026
Union-Closed Sets Conjecture
A deceptively simple frequency conjecture for families closed under union.
Mathematical statement
Every finite nontrivial union-closed family contains an element belonging to at least members.
Statement source: Therefore editorial statement
Statement terms: Source-specific
Editorial wording by Therefore. No statement reuse license is granted.
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
UnionClosed.union_closed
Complete statement target, proof intentionally absentLean 4
theorem union_closed [Nonempty n] (h_ne_singleton_empty : A ≠ {∅}) (h_union_closed : IsUnionClosed A) : ∃ i : n, (1 / 2 : ℚ) * #A ≤ #{x ∈ A | i ∈ x} := 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