Abstract
A family of finite sets is called union-closed if it contains the union of any two sets in it. The Union-Closed Sets Conjecture of Frankl from 1979 states that each union-closed family contains an element that belongs to at least half of the members of the family. In this paper, we study structural properties of union-closed families. It is known that under the inclusion relation, every union-closed family forms a lattice. We call two union-closed families isomorphic if their corresponding lattices are isomorphic. Let \({{\mathcal {F}}}\) be a union-closed family and \(\bigcup _{F\in {\mathcal {F}}} F\) be the universe of \({\mathcal {F}}\). Among all union-closed families isomorphic to \({{\mathcal {F}}}\), we develop an algorithm to find one with a maximum universe, and an algorithm to find one with a minimum universe. We also study properties of these two extremal union-closed families in connection with the Union-Closed Set Conjecture. More specifically, a lower bound of sizes of sets of a minimum counterexample to the dual version of the Union-Closed Sets Conjecture is obtained.
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Guantao Chen: Partially supported by NSF Grant DMS-1855716.
Alexandr Kostochka: Partially supported by NSF Grant DMS-1600592.
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Chen, G., van der Holst, H., Kostochka, A. et al. Extremal Union-Closed Set Families. Graphs and Combinatorics 35, 1495–1502 (2019). https://doi.org/10.1007/s00373-019-02087-2
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DOI: https://doi.org/10.1007/s00373-019-02087-2