Union-closed sets conjecture

Unsolved problem in mathematics:
If any two sets in some finite family of sets have a union that also belongs to the family, must some element belong to at least half of the sets in the family?
<a href="/facts/List_of_unsolved_problems_in_mathematics/RroxGPpP">(more unsolved problems in mathematics)</a>

The union-closed sets conjecture, also known as Frankl’s conjecture, is an <a href="/facts/Open_problem/VuR6yLsW">open problem</a> in <a href="/facts/Combinatorics/k14xUoKT">combinatorics</a> posed by <a href="/facts/P%25C3%25A9ter_Frankl/GpakDRMh">Péter Frankl</a> in 1979. A <a href="/facts/Family_of_sets/WbYof7lP">family of sets</a> is said to be union-closed if the <a href="/facts/Union_(set_theory)/KVVXKfWl">union</a> of any two <a href="/facts/Set_(mathematics)/BfucfHMq">sets</a> from the family belongs to the family. The <a href="/facts/Conjecture/oPWUb7SF">conjecture</a> states: <blockquote> For every finite union-closed family of sets, other than the family containing only the <a href="/facts/Empty_set/ffEk3eA6">empty set</a>, there exists an element that belongs to at least half of the sets in the family.</blockquote>
Professor <a href="/facts/Timothy_Gowers/El3xoEmE">Timothy Gowers</a> 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 <a href="/facts/Mathematical_proof/7ECDrU80">proofs</a> 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 ..."

Union-closed sets conjecture open-in-new

Union-closed sets conjecture