Union-closed families of sets
A family of sets is union-closed if it contains the union of any two of its elements. Reimer (2003)  and Czédli (2009)  investigated the average size of an element of a union-closed family consisting of m subsets of a ground set with n elements. We determine the minimum average size precisely, verifying a conjecture of Czédli, Maróti and Schmidt (2009) . As a consequence, the union-closed conjecture holds if m≥2/3.2 n - in this case some element of [. n] is in at least half the sets of the family. © 2012 Elsevier Inc.
Journal of Combinatorial Theory. Series A
Balla, I., Bollobás, B., & Eccles, T. (2013). Union-closed families of sets. Journal of Combinatorial Theory. Series A, 120 (3), 531-544. https://doi.org/10.1016/j.jcta.2012.10.005