Journal of Combinatorial Theory Series A
All maximum 2-part Sperner families
Journal of Combinatorial Theory Series A
Discrete Mathematics
A note on convex hulls of more-part Sperner families
Journal of Combinatorial Theory Series A
On the structure of maximum 2-part Sperner families
Discrete Mathematics
Sperner theory
Two-Part and k-Sperner Families: New Proofs Using Permutations
SIAM Journal on Discrete Mathematics
All Maximum Size Two-Part Sperner Systems: In Short
Combinatorics, Probability and Computing
Mixed orthogonal arrays, k-dimensional m-part sperner multifamilies, and full multitransversals
Information Theory, Combinatorics, and Search Theory
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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^@?-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 Sperner's theorem, and for the sake of completeness, we give proofs to many simple facts that we use.