Home›Open Questions›Does every finite union-closed family of sets, other than the family holding only the empty set, really contain an element belonging to at least half of the sets in the family?Open QuestionsDoes every finite union-closed family of sets, other than the family holding only the empty set, really contain an element belonging to at least half of the sets in the family?Citation FormatsGeneral ReferenceGeneral Reference Citation TextMathematics Atlas. "Does every finite union-closed family of sets, other than the family holding only the empty set, really contain an element belonging to at least half of the sets in the family?." Accessed August 30, 2026. https://dev-mathematics.interactiveatlas.org/open-questions-index/question-union-closed-sets-half-element-exists.Copy General ReferenceAPA StyleAPA Style Citation TextMathematics Atlas. (n.d.). Does every finite union-closed family of sets, other than the family holding only the empty set, really contain an element belonging to at least half of the sets in the family?. Retrieved August 30, 2026, from https://dev-mathematics.interactiveatlas.org/open-questions-index/question-union-closed-sets-half-element-existsCopy APA StyleBibTeXBibTeX Citation Text@misc{mathematicsatlas-does-every-finite-union-closed-family-of, author = {Mathematics Atlas}, title = {Does every finite union-closed family of sets, other than the family holding only the empty set, really contain an element belonging to at least half of the sets in the family?}, year = {2026}, url = {https://dev-mathematics.interactiveatlas.org/open-questions-index/question-union-closed-sets-half-element-exists}, note = {Accessed August 30, 2026} }Copy BibTeXOpen QuestionCross-Tradition ConnectionsSourcesComments (0)Reader Challenges (0 open reader challenges)Open QuestionThe conjecture is confirmed for families of at most 50 sets and other special cases, and Gilmer's 2022 argument reaches only a weaker constant near 0.38 rather than 0.5, so no proof reaches the full one-half bound for every family.What would resolve this A proof that some element always belongs to at least half of the sets in any union-closed family, or a counterexample family where no element does.OpenCombinatoricsUnion-Closed Sets Conjecture (Wikipedia)Cross-Tradition ConnectionsQuestion OnUnion-Closed Sets Conjecture, Conjectures Well-attested Source Union-Closed Sets Conjecture (Wikipedia)tier 2SourcesUnion-Closed Sets Conjecture (Wikipedia)tier 2Wikipedialead section and Partial results sectionView the SourceComments (0)No comments yet. Be the first to share a thought.Sign in to join the discussion.Reader Challenges (0 open reader challenges)No disputes yet. Spotted an error or a better source? Open the first one.Sign in to dispute this or suggest a correction.View At A Past YearThe atlas records no dated fact of its own for this entry, so there is no other year to choose.Show This Year