An Erd\H{o}s-Ko-Rado theorem for multisets

Let $k$ and $m$ be positive integers. A collection of $k$-multisets from $\{1,\mathrm{.}\mathrm{.}\mathrm{.},m\}$ is intersecting if every pair of multisets from the collection is intersecting. We prove that for $m\ge k+1$, the size of the largest such collection is $\left(\genfrac{}{}{0px}{}{m+k-2}{k-1}\right)$ and that when $m>k+1$, only a collection of all the $k$-multisets containing a fixed element will attain this bound. The size and structure of the largest intersecting collection of $k$-multisets for $m\le k$ is also given.