On averaging Frankl's 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. Frankl’s conjecture from 1979, also known as the union-closed sets conjecture, states that there exists an element a ∈ A with n − 2s(a) ≤ 0. Strengthening a result of Gao and Yu [7] we verify the conjecture for the particular case when m ≥ 3 and n ≥ 2m − 2m/2 . Moreover, for these “large” families F we prove an even stronger version via averaging. Namely, the sum of the n − 2s(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.