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Frankl's union-closed sets conjecture
Conjecture Let 
be a finite family of finite sets, not all empty, that is closed under taking unions. Then there exists 
such that is an element of at least half the members of .
be a finite family of finite sets, not all empty, that is closed under taking unions. Then there exists such that is an element of at least half the members of . This conjecture is notoriously difficult, even though (or should we say `because'?) it involves almost no mathematical structure whatsoever. It was posed by Frankl in the late 1970's. The recent paper of Morris [M] provides a good illustration of the kind of partial results known: Morris extends earlier work to show that the conjecture holds for families containing three 3-subsets of a 5-set, four 3-subsets of a 6-set, or eight 4-subsets of a 6-set. In a different direction, Czédli [C] has proved the conjecture in the case when 
where 
.
Bibliography
[C] G. Czédli, On averaging Frankl's conjecture for large union-closed-sets, J. Combin. Theory Ser. A, to appear.
[M] R. Morris, FC-families and improved bounds for Frankl's conjecture, European J. Combin. 27 (2006), no. 2, 269–282.
[P] B. Poonen, Union-closed families, J. Combin. Theory Ser. A 59 (1992), no. 2, 253–268.
[V] T. P. Vaughan, Three-sets in a union-closed family, J. Combin. Math. Combin. Comput. 49 (2004), 73–84.
[W] P. Wójcik, Union-closed families of sets, Discrete Math. 199 (1999), no. 1–3, 173–182.
* indicates original appearance(s) of problem.
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Frankl's conjecture
For mor einformation and many references see also West's account in: http://www.math.uiuc.edu/~west/openp/unionclos.html