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.
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