On the structure of maximum 2-part Sperner families

AbstractColor the elements of a finite set S with two colors. A collection of subsets of S is called a 2-part Sperner family if whenever for two distinct sets A and B in this collection we have A ⊂ B then B − A has elements of S of both colors. All 2-part Sperner families of maximum size were characterized in Erdős and Katona (1986). In this paper we provide a different, and quite elementary proof of the structure and number of all maximum 2-part Sperner families, using only some elementary properties of symmetric chain decompositions of the poset of all subsets of a finite set