On a weaker notion of cross $t$-intersecting families

We prove that if two families $\mathcal{F}\subseteq \left(\genfrac{}{}{0px}{}{[n]}{k}\right)$ and ${\mathcal{F}}^{\mathrm{\prime }}\subseteq \left(\genfrac{}{}{0px}{}{[n]}{k^{\mathrm{\prime }}}\right)$ satisfy ${\sum }_{1\le i,j\le \mathrm{\ell }}\mid F_i\cap F_j^{\mathrm{\prime }}\mid \ge {\mathrm{\ell }}^{2}t-\mathrm{\ell }+1$ for every choice of distinct $F_1,\dots ,F_{\mathrm{\ell }}\in \mathcal{F}$ and $F_1^{\mathrm{\prime }},\dots ,F_{\mathrm{\ell }}^{\mathrm{\prime }}\in {\mathcal{F}}^{\mathrm{\prime }}$, then $\mid \mathcal{F}\mid \cdot \mid {\mathcal{F}}^{\mathrm{\prime }}\mid \le \left(\genfrac{}{}{0px}{}{n-t}{k-t}\right)\left(\genfrac{}{}{0px}{}{n-t}{k^{\mathrm{\prime }}-t}\right)$, provided that $n$ is sufficiently large. This extends a celebrated theorem of Pyber for large $n$, which determines the tight upper bound for the product of the sizes of cross $1$-intersecting families.