On cross-2-intersecting families

Two families $\mathcal{A}\subseteq \left(\genfrac{}{}{0px}{}{[n]}{k}\right)$ and $\mathcal{B}\subseteq \left(\genfrac{}{}{0px}{}{[n]}{\mathrm{\ell }}\right)$ are called cross-$t$-intersecting if $\mathrm{\mid }A\cap B\mathrm{\mid }\ge t$ for all $A\in \mathcal{A}$, $B\in \mathcal{B}$. Let $n$, $k$ and $\mathrm{\ell }$ be positive integers such that $n\ge 3.38\mathrm{\ell }$ and $\mathrm{\ell }\ge k\ge 2$. In this paper, we will determine the upper bound of $\mathrm{\mid }\mathcal{A}\mathrm{\mid }\mathrm{\mid }\mathcal{B}\mathrm{\mid }$ for cross-$2$-intersecting families $\mathcal{A}\subseteq \left(\genfrac{}{}{0px}{}{[n]}{k}\right)$ and $\mathcal{B}\subseteq \left(\genfrac{}{}{0px}{}{[n]}{\mathrm{\ell }}\right)$. The structures of the extremal families attaining the upper bound are also characterized. The similar result obtained by Tokushige can be considered as a special case of ours when $k=\mathrm{\ell }$, but under a more strong condition $n>3.42k$. Moreover, combined with the results obtained in this paper, the complicated extremal structures attaining the upper bound for nontrivial cases can be relatively easy to reach with similar techniques.