Extremal Problems in Partially Ordered Sets

Book Reviews

“…Some of the best treatments of aspects of (mostly finite) order theory are given by two, now classic books on general combinatorics, by Aigner [1] and Stanley [17], and by Trotter's chapter on partially ordered sets [19]. The surveys provided by West [20,21] in [11,12] are especially interesting for combinatorialists. Readers interested in seeing the broad order theoretic landscape can still profit from Rival's edited volume from the Banff symposium [11] and the proceedings of a number of conferences organized in the 1980s: [10,12,13,14].…”

Maximum-sized antichains in minimal posets

“…B. Nation in Freese [4], which contains also an application of (2) (see also two survey articles [6] and [8]). The following result, which is the converse of (2), has been established recently in [7]:…”

Improved bounds for cross-Sperner systems

“…hence resolving a related conjecture of Hilton (see [3]). Equality is obtained in Seymour's bound precisely when the minimal sets of F are pairwise disjoint from the minimal sets intersecting each set of G. A broad spectrum of research concerning discrete objects with 'Sperner-like' properties have since emerged (see, for example, [1,2,4,5,7,10,11,12,13,19,23]). Many related results concern families satisfying both Sperner-type properties, and additional properties such as conditions on intersections (see, for example [6,15,17,18,20]).…”