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 ± 1 (mod 2k) for almost all odd values of k.