2012/01/20 by Li, P. C., Meagher, Karen
#05B30 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1201.4375
A Sperner k-partition system on a set X is a set of partitions of X into k classes such that the classes of the partitions form a Sperner set system (so no class from a partition is a subset of a class from another partition). These systems were defined by Meagher, Moura and Stevens in \citeMMS who showed that if |X| = k ℓ, then the largest Sperner k-partition system has size (1)/(k)\binom|X|ℓ. In this paper we find bounds on the size of the largest Sperner k-partition system where k does not divide the size of X, specifically, we give an exact bound when k=2 and upper and lower bounds when |X| = 2k+1, |X|=2k+2 and |X| = 3k-1.