A three part Sperner theorem

AbstractKatona has proven a generalization of Sperner's theorem concerning the maximum size of a collection F of ordered triples of elements of three symmetric chain orders, S, T, and U, respectively. It states that if no two elements of F are equal in two components and ordered in the third, and if F satisfies a certain additional condition, then F is maximized by taking all the elements of middle rank. In this paper we give several weaker conditions of F which thus strengthen the theorem, and we provide an easier method of proving it. We obtain results on a related problem about labelling a rectangular array with integer coordinates, from which follows the generalization to symmetric chain orders. We also describe some alternate approaches to the problem and some conjectures