The Structure of Sperner k-families
Journal of Combinatorial Theory, Series A
If P is a partially ordered set, a k-family of P is a subset which contains no chains of length k + 1. This paper examines the structure of the set of k-families of P. An extension of Dilworth's theorem is obtained by relating the maximum size of a k-family to certain partitions of P into chains. A natural lattice ordering on k-families is defined and analyzed, and a number of strong intersection properties are obtained. Finally, thek-families of P are used to define a class of submodular set functions on P, which can be used to generalize a number of results in transversal theory.
Grrene, C. and D. J. Kleitman. “The structure of Sperner k-families”, Journal of Combinatorial Theory 20 (1976), 41-68.