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Graphs and Combinatorics

, Volume 35, Issue 6, pp 1495–1502 | Cite as

Extremal Union-Closed Set Families

  • Guantao ChenEmail author
  • Hein van der Holst
  • Alexandr Kostochka
  • Nana Li
Original Paper
  • 28 Downloads

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.

Keywords

Family of sets Union-closed sets Normal and irreducible families 

Notes

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Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  • Guantao Chen
    • 1
    Email author
  • Hein van der Holst
    • 1
  • Alexandr Kostochka
    • 2
    • 3
  • Nana Li
    • 4
  1. 1.Department of Mathematics and StatisticsGeorgia State UniversityAtlantaUSA
  2. 2.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  3. 3.Sobolev Institute of MathematicsNovosibirskRussia
  4. 4.Department of Computing ScienceUmeå UniversityUmeåSweden

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