An upper bound on the size of diamond-free families of sets

Let La(n, P) be the maximum size of a family of subsets of [n] = {1, 2,..., n} not containing P as a (weak) subposet. The diamond poset, denoted Q(2), is defined on four elements x, y, z, w with the relations x < y, z and y, z < w. La(n, P) has been studied for many posets; one of the major open problems is determining La(n, Q(2)). It is conjectured that La(n, Q(2)) = (2 + o(1))(n/left perpendicular n/2 right perpendicular), and infinitely many significantly different, asymptotically tight constructions are known. Studying the average number of sets from a family of subsets of [n] on a maximal chain in the Boolean lattice 2([n]) has been a fruitful method. We use a partitioning of the maximal chains and introduce an induction method to show that La(n, Q(2)) <= (2.20711 + o(1))(n/left perpendicular n/2 right perpendicular), improving on the earlier bound of (2.25 + o(1))(n/left perpendicular n/2 right perpendicular) by Kramer, Martin and Young. (C) 2018 Elsevier Inc. All rights reserved.