On nontrivial cross-t-intersecting families

Two families $A\subseteq \left(\begin{array}{c}\left[n\right]\\ k\end{array}\right)$ and $B\subseteq \left(\begin{array}{c}\left[n\right]\\ \ell \end{array}\right)$ are called nontrivial cross-t-intersecting if $|A\cap B|\ge t$ for all $A\in A$, $B\in B$ and $\left|{\bigcap }_{A\in A\cup B}A\right|<t$. In this paper we will determine the upper bound of $\left|A\right|\left|B\right|$ for nontrivial cross-t-intersecting families $A\subseteq \left(\begin{array}{c}\left[n\right]\\ k\end{array}\right)$ and $B\subseteq \left(\begin{array}{c}\left[n\right]\\ \ell \end{array}\right)$ for positive integers n, k, ℓ and t such that $n\ge \max \left\{\right(t+1\left)\right(k-t+1),(t+1\left)\right(\ell -t+1\left)\right\}$ and $t\ge 3$. The structures of the extremal families attaining the upper bound are also characterized. As a byproduct of the main result in this paper, one product version of Erdős–Ko–Rado Theorem for two families of cross-t-intersecting can be easily obtained which gives a confirmative answer to one conjecture by Tokushige.