On the Scope of Averaging for Frankl’s Conjecture

Abstract

Let $\mathcal F$ be a union-closed family of subsets of an m-element set A. Let $n=|{\mathcal F}|\ge 2$. For b ∈ A let w(b) denote the number of sets in $\mathcal F$ containing b minus the number of sets in $\mathcal F$ 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. $\mathcal F$ is said to satisfy the averaged Frankl’s property if this average is non-negative. Although this much stronger property does not hold for all union-closed families, the first author (Czédli, J Comb Theory, Ser A, 2008) verified the averaged Frankl’s property whenever n ≥ 2^m − 2^m/2 and m ≥ 3. The main result of this paper shows that (1) we cannot replace 2^m/2 with the upper integer part of 2^m/3, and (2) if Frankl’s conjecture is true (at least for m-element base sets) and $n\ge 2^m-\lfloor 2^m/3\rfloor$ then the averaged Frankl’s property holds (i.e., 2^m/2 can be replaced with the lower integer part of 2^m/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.