Analogues of Katona's and Milner's Theorems for two families

Let $n>s>0$ be integers, $X$ an $n$-element set and $\mathscr{A}, \mathscr{B}\subset 2^X$twofamilies\mathrm{.}If$|A\cup B|\le s$forall$A\in\mathscr{A}, B\in \mathscr{B}$,then$\mathscr{A}$and$\mathscr{B}$ are called cross $s$-union. Assuming that neither $\mathcal{A}$ nor $\mathcal{B}$ is empty, we prove several best possible bounds. In particular, we show that $\mathrm{\mid }\mathcal{A}\mathrm{\mid }+\mathrm{\mid }\mathcal{B}\mathrm{\mid }\le 1+\sum _{0\le i\le s}\left(\genfrac{}{}{0px}{}{n}{i}\right)$. Supposing $n\ge 2s$ and $\mathcal{A},\mathcal{B}$ are antichains, we show that $\mathrm{\mid }\mathcal{A}\mathrm{\mid }+\mathrm{\mid }\mathcal{B}\mathrm{\mid }\le \left(\genfrac{}{}{0px}{}{n}{1}\right)+\left(\genfrac{}{}{0px}{}{n}{s-1}\right)$ unless $\mathcal{A}=\{\mathrm{\varnothing }\}$ or $\mathcal{B}=\{\mathrm{\varnothing }\}$. An analogous result for three families is established as well.