On the Scope of Averaging for Frankl's Conjecture

Let F be a union-closed family of subsets of an m-element set A. Let n = |F| ≥ 2. For b ∈ A let w(b) denote the number of sets in F containing b minus the number of sets in F not containing b. Frankl’s conjecture from 1979, also known as the union-closed sets conjecture, states that there exists an element b ∈ A with w(b) ≥ 0. The present paper deals with the average of the w(b), computed over all b ∈ A. F is said to satisfy the averaged Frankl’s property if this average is nonnegative. Although this much stronger property does not hold for all unionclosed families, the first author [7] verified the averaged Frankl’s property whenever n ≥ 2m − 2m/2 and m ≥ 3. The main result of this paper shows that (1) we cannot replace 2m/2 with the upper integer part of 2m/3, and (2) if Frankl’s conjecture is true (at least for m-element base sets) and n ≥ 2m − b2m/3c then the averaged Frankl’s property holds (i.e., 2m/2 can be replacedwith the lower integer part of 2m/3). The proof combines elementary facts from combinatorics and lattice theory. The paper is self-contained, and the reader is assumed to be familiar neither with lattices nor with combinatorics.