The maximum product of sizes of cross-$t$-intersecting families

Two families of sets $\mathcal{A}$ and $\mathcal{B}$ are called \emph{cross-$t$-intersecting} if $\mathrm{\mid }A\cap B\mathrm{\mid }\ge t$ for all $A\in \mathcal{A}$ and $B\in \mathcal{B}$. Determining the maximum product of sizes for such cross-$t$-intersecting families is an active problem in extremal set theory. In this paper, we verify the following cross-$t$-intersecting version of the Erdős-Ko-Rado theorem: For $k\ge l\ge t\ge 3$ and $\min \{m,n\}\ge (t+1)(k-t+1)$, the maximun value of $\mathrm{\mid }\mathcal{A}\mathrm{\mid }\mathrm{\mid }\mathcal{B}\mathrm{\mid }$ for two cross-$t$-intersecting families $\mathcal{A}\subseteq \left(\genfrac{}{}{0px}{}{[n]}{k}\right)$ and $\mathcal{B}\subseteq \left(\genfrac{}{}{0px}{}{[m]}{l}\right)$ is $\left(\genfrac{}{}{0px}{}{n-t}{k-t}\right)\left(\genfrac{}{}{0px}{}{m-t}{l-t}\right)$. Moreover, we characterize the extremal families attaining the upper bound. Our result confirms a conjecture of Tokushige for $t\ge 3$, and actually proves a more general result.