Union-closed sets conjecture
|If any two sets in some 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?|
In combinatorial mathematics, the union-closed sets conjecture is an elementary problem, posed by Péter Frankl in 1979 and still open. A family of sets is said to be union-closed if the union of any two sets from the family remains in the family. The conjecture states that for any finite union-closed family of finite sets, other than the family consisting only of the empty set, there exists an element that belongs to at least half of the sets in the family.
If F is a union-closed family of sets, the family of complement sets to sets in F is closed under intersection, and an element that belongs to at least half of the sets of F belongs to at most half of the complement sets. Thus, an equivalent form of the conjecture (the form in which it was originally stated) is that, for any intersection-closed family of sets that contains more than one set, there exists an element that belongs to at most half of the sets in the family.
Although stated above in terms of families of sets, Frankl's conjecture has also been formulated and studied as a question in lattice theory. A lattice is a partially ordered set in which for two elements x and y there is a unique greatest element less than or equal to both of them (the meet of x and y) and a unique least element greater than or equal to both of them (the join of x and y). The family of all subsets of a set S, ordered by set inclusion, forms a lattice in which the meet is represented by the set-theoretic intersection and the join is represented by the set-theoretic union; a lattice formed in this way is called a Boolean lattice. The lattice-theoretic version of Frankl's conjecture is that in any finite lattice there exists an element x that is not the join of any two smaller elements, and such that the number of elements greater than or equal to x totals at most half the lattice, with equality only if the lattice is a Boolean lattice. As Abe (2000) shows, this statement about lattices is equivalent to the Frankl conjecture for union-closed sets: each lattice can be translated into a union-closed set family, and each union-closed set family can be translated into a lattice, such that the truth of the Frankl conjecture for the translated object implies the truth of the conjecture for the original object. This lattice-theoretic version of the conjecture is known to be true for several natural subclasses of lattices (Abe 2000; Poonen 1992; Reinhold 2000) but remains open in the general case.
Families known to satisfy the conjecture
The conjecture has been proven for many special cases of union-closed set families. In particular, it is known to be true for
- families of at most 46 sets (Roberts & Simpson 2010).
- families of sets such that their union has at most 11 elements, and
- families of sets in which the smallest set has one or two elements.
Péter Frankl stated the conjecture, in terms of intersection-closed set families, in 1979, and so the conjecture is usually credited to him and sometimes called the Frankl conjecture. The earliest publication of the union-closed version of the conjecture appears to be by Duffus (1985).
- Bošnjak and Marković (2008), improving previous bounds by Morris (2006), Lo Faro (1994) and others.
- Sarvate and Renaud (1989), since rediscovered by several other authors. If a one-element or two-element set S exists, some element of S belongs to at least half the sets in the family, but the same property does not hold for three-element sets, due to counterexamples of Sarvate, Renaud, and Ronald Graham.
- Abe, Tetsuya (2000). "Strong semimodular lattices and Frankl's conjecture". Algebra Universalis 44 (3–4): 379–382. doi:10.1007/s000120050195.
- Bošnjak, Ivica; Marković, Peter (2008). "The 11-element case of Frankl's conjecture". Electronic Journal of Combinatorics 15 (1): R88.
- Duffus, D. (1985). "Open problem session". In Rival, I.. Graphs and Order. D. Reidel. p. 525.
- Lo Faro, Giovanni (1994). "Union-closed sets conjecture: improved bounds". J. Combin. Math. Combin. Comput. 16: 97–102. MR 1301213.
- Morris, Robert (2006). "FC-families and improved bounds for Frankl's conjecture". European Journal of Combinatorics 27 (2): 269–282. doi:10.1016/j.ejc.2004.07.012. MR 2199779.
- Poonen, Bjorn (1992). "Union-closed families". Journal of Combinatorial Theory, Series A 59 (2): 253–268. doi:10.1016/0097-3165(92)90068-6. MR 1149898.
- Reinhold, Jürgen (2000). "Frankl's conjecture is true for lower semimodular lattices". Graphs Combin. 16 (1): 115–116. doi:10.1007/s003730050008.
- Roberts, Ian; Simpson, Jamie (2010). "A note on the union-closed sets conjecture". Australas. J. Combin. 47: 265–267.
- Sarvate, D. G.; Renaud, J.-C. (1989). "On the union-closed sets conjecture". Ars Combin. 27: 149–153. MR 0989460.