New bounds on the maximum size of Sperner partition systems. (December 2020) - Chang, Yanxun; Colbourn, Charles J.; Gowty, Adam; Horsley, Daniel; Zhou, Junling
Abstract: An ( n, k ) - Sperner partition system is a collection of partitions of some n -set, each into k nonempty classes, such that no class of any partition is a subset of a class of any other. The maximum number of partitions in an ( n, k ) -Sperner partition system is denoted SP ( n, k ) . In this paper we introduce a new construction for Sperner partition systems and use it to asymptotically determine SP ( n, k ) in many cases as n k becomes large. We also give a slightly improved upper bound for SP ( n, k ) and exhibit an infinite family of parameter sets ( n, k ) for which this bound is tight.