Sharp results concerning disjoint cross-intersecting families

For an $n$-element set $X$ let $\left(\genfrac{}{}{0px}{}{X}{k}\right)$ be the collection of all its $k$-subsets. Two families of sets $\mathcal{A}$ and $\mathcal{B}$ are called cross-intersecting if $A\cap B\mathrm{\ne }\mathrm{\varnothing }$ holds for all $A\in \mathcal{A}$, $B\in \mathcal{B}$. Let $f(n,k)$ denote the maximum of $\min{|\mathcal A|, |\mathcal B|}$ where the maximum is taken over all pairs of {\em disjoint}, cross-intersecting families $\mathcal{A},\mathcal{B}\subset \left(\genfrac{}{}{0px}{}{[n]}{k}\right)$. Let $c={\log }_{2}e$. We prove that $f(n,k)=\lfloor \frac{1}{2}\left(\genfrac{}{}{0px}{}{n-1}{k-1}\right)\rfloor$ essentially iff $n>c{k}^2$ (cf. Theorem~1.4 for the exact statement). Let $f^{\ast }(n,k)$ denote the same maximum under the additional restriction that the intersection of all members of both $\mathcal{A}$ and $\mathcal{B}$ are empty. For $k\ge 5$ and $n\ge k^3$weshowthat$f^*(n,k)=\left\lfloor\frac12\left(\binom{n-1}{k-1}-\binom{n-2k}{k-1}\right)\right\rfloor+1$andtherestrictionon$n$ is essentially sharp (cf. Theorem~5.4).