On averaging Frankl's conjecture for large union-closed-sets

Abstract: Let be a union-closed family of subsets of an m-element set A. Let and for let denote the number of sets in that contain a. Frankl''s conjecture from 1979, also known as the union-closed sets conjecture, states that there exists an element with . 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 and . Moreover, for these “large” families we prove an even stronger version via averaging. Namely, the sum of the , for all , 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. [Copyright &y& Elsevier]