Maximum Antichains of Rectangular Arrays

Given a finite partially ordered set B, it is often of interest to describe the largest possible subsets SC B which consist of mutually incomparable elements of B. Such subsets are usually called maximum antichains or maximum Sperner families of .P after Sperner [I 51, who first characterized these maximum antichains in the case that 9 is a finite Boolean lattice, i.e., the set of all subsets of some finite set partially ordered by inclusion. Recently, great advances have been made for questions of this type by Canfield, Greene, Harper, Katona, Kleitman, Lubell, Milne and others (e.g., see [I, 5-9, 11, 121) In particular, Canfield [l-3], has recently settled in the negative the old question of Rota [13, 141: