Cross-Sperner families

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