More constructions for Sperner partition systems

An (𝑛,𝑘)-Sperner partition system is a set of partitions of some 𝑛n-set such that each partition has 𝑘k 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 𝑛𝑘 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(mod2𝑘) for almost all odd values of 𝑘.