On a conjecture of Tokushige for cross-$t$-intersecting families

Two families of sets $\mathcal{A}$ and $\mathcal{B}$ are called cross-$t$-intersecting if $\mathrm{\mid }A\cap B\mathrm{\mid }\ge t$ for all $A\in \mathcal{A}$, $B\in \mathcal{B}$. An active problem in extremal set theory is to determine the maximum product of sizes of cross-$t$-intersecting families. This incorporates the classical Erdős–Ko–Rado (EKR) problem. In the present paper, we prove that if $\mathcal{A}$ and $\mathcal{B}$ are cross-$t$-intersecting families of $\left(\genfrac{}{}{0px}{}{[n]}{k}\right)$ with $k\ge t\ge 3$ and $n\ge (t+1)(k-t+1)$, then $\mathrm{\mid }\mathcal{A}\mathrm{\mid }\mathrm{\mid }\mathcal{B}\mathrm{\mid }\le {\left(\genfrac{}{}{0px}{}{n-t}{k-t}\right)}^2$; moreover, if $n>(t+1)(k-t+1)$, then equality holds if and only if $\mathcal{A}=\mathcal{B}$ is a maximum $t$-intersecting subfamily of $\left(\genfrac{}{}{0px}{}{[n]}{k}\right)$. This confirms a conjecture of Tokushige for $t\ge 3$.