A Sperner-Type Theorem for Set-Partition Systems - Meagher, Karen; Moura, Lucia; Stevens, Brett (The Electronic Journal of Combinatorics, 2005)
A Sperner partition system is a system of set partitions such that any two set partitions P and Q in the system have the property that for all classes A of P and all classes B of Q, A otB and B otA. A k-partition is a set partition with k classes and a k-partition is said to be uniform if every class has the same cardinality c=n/k. In this paper, we prove a higher order generalization of Sperner’s Theorem. In particular, we show that if k divides n the largest Sperner k-partition system on an n-set has cardinality n-1 n/k-1 and is a uniform partition system. We give a bound on the cardinality of a Sperner k-partition system of an n-set for any k and n.