A tour of M-part L-Sperner families

In this paper we investigate common generalizations of more-part and L-Sperner families. We prove a BLYM inequality for M-part L-Sperner families and obtain results regarding the homogeneity of such families of maximum size through the convex hull method. We characterize those M-part Sperner problems, where the maximum family size is the classical ( n ⌊ n / 2 ⌋ ) . We make a conjecture on the maximum size of M-part Sperner families for the case of equal parts of size 2 ℓ − 1 and prove the conjecture in some special cases. We introduce the notion of k-fold M-part Sperner families, which specializes to the concept of M-part Sperner families for k = 1 , and generalize some M-part Sperner results to k-fold M-part Sperner families. We also approach the M-part Sperner problem from the viewpoints of graph product and linear programming, and prove the 2-part Sperner theorem using linear programming. This paper can be used as a survey, as in addition to the new results, problems and conjectures, we provide a number of alternative proofs, discuss at length a number of generalizations of Spernerʹs theorem, and for the sake of completeness, we give proofs to many simple facts that we use.