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

A central problem in extremal set theory is to determine or estimate $m(n,k,t),n>2k\geq 2t$, the maximum size of an intersecting $k$-graph and covering number at least $t$(see the paper for the definitions). For $t=1$ and $2$ the classical Erd\H{o}s-Ko-Rado Theorem and the Hilton-Milner Theorem provide the answer.The complete solution for $t=3$ was only achieved recently . There are some partial results for $t=4,5$ but for the general case even to determine the asymptotic appears to be hopelessly difficult . Denoting by $\widetilde{m}(n,k,t)$ the maximum of $|\mathcal{F}||\mathcal{G}|$ for a pair of cross-intersecting $k$-graphs with covering number at least $t$, $\widetilde{m}(n,k,t)\geq {m}(n,k,t)^2$ is obvious. Pyber showed that equality holds for $t=1$. The same was shown for $t=2$ in a wide range(cf.[7]). Quite surprisingly our results show that the inequality is strict for $t\geq 3$ and for $n>n_0(k,t)$, Theorem 1.7 determines the exact value of $\widetilde{m}(n,k,t)$ for $k>2t$ and $n$ sufficiently large.