  A pair of families $(\cF,\cG)$ is said to be \emph{cross-Sperner} if there
exists no pair of sets $F \in \cF, G \in \cG$ with $F \subseteq G$ or $G
\subseteq F$. There are two ways to measure the size of the pair $(\cF,\cG)$:
with the sum $|\cF|+|\cG|$ or with the product $|\cF|\cdot |\cG|$. We show that
if $\cF, \cG \subseteq 2^{[n]}$, then $|\cF||\cG| \le 2^{2n-4}$ and
$|\cF|+|\cG|$ is maximal if $\cF$ or $\cG$ consists of exactly one set of size
$\lceil n/2 \rceil$ provided the size of the ground set $n$ is large enough and
both $\cF$ and $\cG$ are non-empty.
