On Cross-intersecting Sperner Families

Two sets $\mathscr{A}$ and $\mathscr{B}$ are said to be cross-intersecting if $X\cap Y\neq\emptyset$ for all $X\in\mathscr{A}$ and $Y\in\mathscr{B}$. Given two cross-intersecting Sperner families (or antichains) $\mathscr{A}$ and $\mathscr{B}$ of $\mathbb{N}_n$, we prove that $|\mathscr{A}|+|\mathscr{B}|\le 2{{n}\choose{\lceil{n/2}\rceil}}$ if $n$ is odd, and $|\mathscr{A}|+|\mathscr{B}|\le {{n}\choose{n/2}}+{{n}\choose{(n/2)+1}}$ if $n$ is even. Furthermore, all extremal and almost-extremal families for $\mathscr{A}$ and $\mathscr{B}$ are determined.