$s$-almost $t$-intersecting families for finite sets

A family $\mathcal{F}$ of $k$-subsets of an $n$-set is called $s$-almost $t$-intersecting if each member is $t$-disjoint with at most $s$ members. In this paper, we prove that, if $\mid \mathcal{F}\mid$ is maximum, then $\mathcal{F}$ consists of all $k$-subsets containing a fixed $t$-subset. Consequently, it is natural to consider the maximum-sized $\mathcal{F}$ with $\mid {\bigcap }_{F\in \mathcal{F}}F\mid <t$. The famous Hilton-Milner theorem settles the case where $\mathcal{F}$ is $t$-intersecting. We characterize the remaining case completely.