Math Conjectures

Union-Closed Sets Conjecture (Frankl's Conjecture)

OPENLandmarkConjectureProposed 1979 · Standard version

Canonical statement

Let F\mathcal F be a finite, nonempty family of finite sets such that A∪B∈FA\cup B\in\mathcal F for all A,B∈FA,B\in\mathcal F, and assume ⋃F≠∅\bigcup\mathcal F\ne\varnothing. Then some element x∈⋃Fx\in\bigcup\mathcal F belongs to at least ∣F∣/2\lvert\mathcal F\rvert/2 members of F\mathcal F.
View source LaTeX
Let \(\mathcal F\) be a finite, nonempty family of finite
sets such that \(A\cup B\in\mathcal F\) for all
\(A,B\in\mathcal F\), and assume \(\bigcup\mathcal F\ne\varnothing\).
Then some element \(x\in\bigcup\mathcal F\) belongs to at least
\(\lvert\mathcal F\rvert/2\) members of \(\mathcal F\).
COMB-002:[BruhnSchaudt2015][AlweissHuangSellke2024]
Combinatorics

Notes

Frankl's conjecture, dating from 1979, concerns finite families of finite sets closed under union: if A,B∈FA,B\in\mathcal F implies A∪B∈FA\cup B\in\mathcal F and F\mathcal F contains a nonempty set, then some element of the ground set should lie in at least half of the members of F\mathcal F. Despite its innocuous appearance the problem has resisted a wide range of techniques; its history and the many partial approaches are surveyed by Bruhn and Schaudt [BruhnSchaudt2015].

The conjecture has been verified in numerous special cases, such as families over small ground sets or with structural restrictions. A recent breakthrough came from an information-theoretic method that yields a universal constant: every union-closed family has an element belonging to at least a fixed fraction of its sets. The bound (3−5)/2≈0.38(3-\sqrt5)/2\approx0.38 was established along these lines by Alweiss, Huang and Sellke [AlweissHuangSellke2024].

The gap between (3−5)/2(3-\sqrt5)/2 and the conjectured sharp value 1/21/2 is exactly what remains, and closing it may require ideas beyond the present arguments; the conjecture is open.

References (2)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.