Two families $\mathcal{ F}$ and $\mathcal{ G}$ of $k$-subsets of an $n$-set are called $s$-almost cross-$t$-intersecting if each member in $\mathcal{ F}$ (resp. $\mathcal{ G}$) is $t$-disjoint with at most $s$ members in $\mathcal{ G}$ (resp. $\mathcal{ F}$). In this paper, we characterize the $s$-almost cross-$t$-intersecting families with the maximum product of their sizes. Furthermore, we provide a corresponding stability result after studying the $s$-almost cross-$t$-intersecting families which are not cross-$t$-intersecting.
Categories: math.CO