More constructions for Sperner partition systems. Issue 9 (4th June 2021) - Gowty, Adam; Horsley, Daniel
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 2 k ) for almost all odd values of k .