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) [16] and Czédli (2009) [2] 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) [3]. 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.