Chen, Guantao; van der Holst, Hein; Kostochka, Alexandr; Li, Nana Extremal union-closed set families. (English) Zbl 1431.05141 Graphs Comb. 35, No. 6, 1495-1502 (2019). Summary: 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 (see [P. Frankl, in: Handbook of combinatorics. Vol. 1–2. Amsterdam: Elsevier (North-Holland); Cambridge, MA: MIT Press. 1293–1329 (1995; Zbl 0844.05094)]) 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. MSC: 05D05 Extremal set theory Keywords:family of sets; union-closed sets; normal and irreducible families Citations:Zbl 0844.05094 PDFBibTeX XMLCite \textit{G. Chen} et al., Graphs Comb. 35, No. 6, 1495--1502 (2019; Zbl 1431.05141) Full Text: DOI References: [1] Balla, Igor; Bollobás, Béla; Eccles, Tom, Union-closed families of sets, J. Combin. Theory Ser. A, 120, 531-544 (2013) · Zbl 1259.05177 [2] Bruhn, Henning; Charbit, Pierre; Schaudt, Oliver; Telle, Jan Arne, The graph formulation of the union-closed sets conjecture, Eur. J. Combin., 43, 210-219 (2015) · Zbl 1301.05183 [3] Bruhn, H., Schaudt, O.: The union-closed sets conjecture almost holds for almost all random bipartite graphs. In: The 7th European Conference on Combinatorics, Graph Theory and Applications, volume 16 of CRM Series, pp. 79-84. Ed. Norm., Pisa, (2013) · Zbl 1293.05352 [4] Bruhn, H.; Schaudt, O., The journey of the union-closed sets conjecture, Graphs Combin., 31, 2043-2074 (2015) · Zbl 1327.05249 [5] Czédli, G.; Maróti, M.; Schmidt, ET, On the scope of averaging for Frankl’s conjecture, Order, 26, 31-48 (2009) · Zbl 1229.05259 [6] Falgas-Ravry, V., Minimal weight in union-closed families, Electron. J. Combin., 18, 95 (2011) · Zbl 1220.05127 [7] Frankl, P.: The Handbook of Combinatorics. MIT Press, Cambridge (1995). chapter 24 [8] Knill, E.: Graph generated union-closed families of sets (1994), arXiv:math/9409215v1 [math.CO] [9] Reimer, D., An average set size theorem, Combin. Probab. Comput., 12, 89-93 (2003) · Zbl 1013.05083 [10] Roberts, I.; Simpson, J., A note on the union-closed sets conjecture, Aust. J. Combin., 47, 265-267 (2010) · Zbl 1277.05161 [11] Wójcik, P., Union-closed families of sets, Discr. Math., 199, 173-182 (1999) · Zbl 0927.05081 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.