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Let $m$, $n$, and $k$ be integers satisfying $0 < k \leq n < 2k \leq m$. Afamily of sets $\mathcal{F}$ is called an $(m,n,k)$-intersecting family if$\binom{[n]}{k} \subseteq \mathcal{F} \subseteq \binom{[m]}{k}$ and any pair ofmembers of $\mathcal{F}$ have nonempty intersection. Maximum $(m,k,k)$- and$(m,k+1,k)$-intersecting families are determined by the theorems ofErd\H{o}s-Ko-Rado and Hilton-Milner, respectively. We determine the maximumfamilies for the cases $n = 2k-1, 2k-2, 2k-3$, and $m$ sufficiently large.

A Common Generalization of the Theorems of Erd\H{o}s-Ko-Rado and Hilton-Milner