On the product of cross-intersecting families with maximal covering number

For integers $k ,\ell \geq 2$ let $m(k,\ell)$ denote the maximum of $|\mathcal{F}| |\mathcal{G}|$ where the maximum is taken over all pairs of cross-intersecting families, $\mathcal{F}$ being a $k$-graph with covering number $\ell$ and $\mathcal{G}$ a $\ell$-graph with covering number $k$ (see the paper for the definitions). Erdos and Lovasz initiated the study of the one family version. That is, they provided lower and upper bounds on the maximal size $m(k)=|\mathcal{F}|$ where $\mathcal{F}$ is an intersecting k-graph with covering number $k$. In many similar situations $m(k,k)=m(k)^2$ holds. However, as our results show $m(k,k)/m(k)^2$ is tending to infinity as $k$ grows(Th.1.5) . For $k>k_0$ we establish the exact value $m(k,k)=(k^{k-1}+k-1)^2$(Th.1.6). As to smaller values we prove $m(3,3)=121$ (Th.1.7) and determine $m(2,k) $ for all $k\geq 2$ (Th.1.8).