The union-closed sets conjecture
A finite family of finite sets is union-closed if the union of any two members is again a member. Prove that every finite union-closed family that contains at least one nonempty set has an element lying in at least half of the sets of the family.
History and references
Still open as of 2026.
Péter Frankl stated the conjecture in 1979 in the equivalent language of intersection-closed families, so it is often called Frankl's conjecture. Timothy Gowers has called it one of the best known open problems in combinatorics. For decades nobody could even show that some element lies in a fixed positive fraction of the sets. In 2022 Justin Gilmer gave an information-theoretic argument showing that some element lies in at least 1% of them. Within days other groups improved the constant to about 0.38, and the best known value is now about 0.3827. Reaching one half seems to need a new idea.
References
Solutions 0
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