  A collection of families $(\mathcal{F}_{1}, \mathcal{F}_{2} , \cdots ,
\mathcal{F}_{k}) \in \mathcal{P}([n])^k$ is cross-Sperner if there is no pair
$i \not= j$ for which some $F_i \in \mathcal{F}_i$ is comparable to some $F_j
\in \mathcal{F}_j$. Two natural measures of the `size' of such a family are the
sum $\sum_{i = 1}^k |\mathcal{F}_i|$ and the product $\prod_{i = 1}^k
|\mathcal{F}_i|$. We prove new upper and lower bounds on both of these measures
for general $n$ and $k \ge 2$ which improve considerably on the previous best
bounds. In particular, we construct a rich family of counterexamples to a
conjecture of Gerbner, Lemons, Palmer, Patk\'{o}s, and Sz\'{e}csi from 2011.
