On averaging Franklʹs conjecture for large union-closed-sets

Let F be a union-closed family of subsets of an m-element set A. Let n = | F | ⩾ 2 and for a ∈ A let s ( a ) denote the number of sets in F that contain a. Franklʹs conjecture from 1979, also known as the union-closed sets conjecture, states that there exists an element a ∈ A with n − 2 s ( a ) ⩽ 0 . Strengthening a result of Gao and Yu [W. Gao, H. Yu, Note on the union-closed sets conjecture, Ars Combin. 49 (1998) 280–288] we verify the conjecture for the particular case when m ⩾ 3 and n ⩾ 2 m − 2 m / 2 . Moreover, for these “large” families F we prove an even stronger version via averaging. Namely, the sum of the n − 2 s ( a ) , for all a ∈ A , is shown to be non-positive. Notice that this stronger version does not hold for all union-closed families; however we conjecture that it holds for a much wider class of families than considered here. Although the proof of the result is based on elementary lattice theory, the paper is self-contained and the reader is not assumed to be familiar with lattices.