The union-closed sets conjecture is an open problem in combinatorics posed by Péter Frankl in 1979. A family of sets is said to be union-closed if the union of any two sets from the family belongs to the family. The conjecture states: For every finite union-closed family of sets, other than the family containing only the empty set, there exists an element that belongs to at least half of the sets in the family. Professor Timothy Gowers has called this "one of the best known open problems in combinatorics" and has said that the conjecture "feels as though it ought to be easy (and as a result has attracted a lot of false proofs over the years). A good way to understand why it isn't easy is to spend an afternoon trying to prove it. That clever averaging argument you had in mind doesn't work ..." Example The family of setsconsists of five different sets and is union-closed. The element is contained in three of the five sets (and so is the element ), thus the conjecture holds in this case. Basic results It is easy to show that if a union-closed family contains a singleton (as in the example above), then the element must occur in at least half of the sets of the family. If there is a counterexample to the conjecture, then there is also a counterexample consisting only of finite sets. Therefore, without loss of generality, we will assume that all sets in the given union-closed family are finite. Given a finite non-empty set , the power set consisting of all subsets of is union-closed. Each element of is contained in exactly half of the subsets of . Therefore, in general we cannot ask for an element contained in more than half of the sets of the family: the bound of the conjecture is sharp. Equivalent forms Intersection formulation The union-closed set conjecture is true if and only if a set system which is intersection-closed contains an element of in at most half of the sets of , where is the universe set, i.e. the union of all members of the system . The following facts show the equivalence. Firstly, we show that a set system is union-closed if and only if its complement is intersection-closed. Lemma 1. If is a union-closed family of sets with universe , the family of complement sets to sets in