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

Two families of sets $A$ and $B$ are called cross-t-intersecting if $|A\cap B|\ge t$ for all $A\in A$, $B\in B$. 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 $A\subseteq \left(\begin{array}{c}\left[n\right]\\ k\end{array}\right)$ and $B\subseteq \left(\begin{array}{c}\left[n\right]\\ k\end{array}\right)$ are cross-t-intersecting with $k\ge t\ge 3$ and $n\ge (t+1)(k-t+1)$, then $\left|A\right|\left|B\right|\le {\left(\begin{array}{c}n-t\\ k-t\end{array}\right)}^2$. Moreover, equality holds if and only if $A=B$ is a maximum t-intersecting subfamily of $\left(\begin{array}{c}\left[n\right]\\ k\end{array}\right)$. This confirms a conjecture of Tokushige for $t\ge 3$.