On Cross-intersecting Sperner Families

Two sets $\mathcal{A}$ and $\mathcal{B}$ are said to be cross-intersecting if $X\cap Y\mathrm{\ne }\mathrm{\varnothing }$ for all $X\in \mathcal{A}$ and $Y\in \mathcal{B}$. Given two cross-intersecting Sperner families (or antichains) $\mathcal{A}$ and $\mathcal{B}$ of ${\mathbb{N}}_n$, we prove that $|\mathscr{A}|+|\mathscr{B}|\le 2{{n}\choose{\lceil{n/2}\rceil}}$if$n$isodd,and$|\mathscr{A}|+|\mathscr{B}|\le {{n}\choose{n/2}}+{{n}\choose{(n/2)+1}}$if$n$iseven\mathrm{.}Furthermore,allextremalandalmost-extremalfamiliesfor$\mathscr{A}$and$\mathscr{B}$ are determined.