Sperner partition systems

A \textsl{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 \cite{MMS} who showed that if $\mathrm{\mid }X\mathrm{\mid }=k\mathrm{\ell }$, then the largest Sperner $k$-partition system has size $\frac{1}{k}\left(\genfrac{}{}{0px}{}{\mathrm{\mid }X\mathrm{\mid }}{\mathrm{\ell }}\right)$. 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 $\mathrm{\mid }X\mathrm{\mid }=2k+1$, $\mathrm{\mid }X\mathrm{\mid }=2k+2$ and $|X| = 3k-1$.