More constructions for Sperner partition systems

Abstract

An ( n , k )‐Sperner partition system is a set of partitions of some n‐set such that each partition has k nonempty parts and no part in any partition is a subset of a part in a different partition. 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 based on a division of the ground set into many equal‐sized parts. We use this to asymptotically determine SP ( n , k ) in many cases where n k is bounded as n becomes large. Further, we show that this construction produces a Sperner partition system of maximum size for numerous small parameter sets ( n , k ). By extending a separate existing construction, we also establish the asymptotics of SP ( n , k ) when n ≡ k ± 10.3em ( mod0.3em 2 k ) for almost all odd values of k.