Erdős-Ko-Rado theorem for bounded multisets

Let $k,m,n$ be positive integers with $k⩾2$. A k-multiset of ${\left[n\right]}_m$ is a collection of k integers from the set $\{1,2,\dots ,n\}$ in which the integers can appear more than once but at most m times. A family of such k-multisets is called an intersecting family if every pair of k-multisets from the family have non-empty intersection. A finite sequence of real numbers $(a_1,a_2,\dots ,a_n)$ is said to be unimodal if there is some $k\in \{1,2,\dots ,n\}$, such that $a_1⩽a_2⩽\dots ⩽a_{k-1}⩽a_k⩾a_{k+1}⩾\dots ⩾a_n$. Given $m,n,k$, denote $C_{k,\ell }$ as the coefficient of $x^k$ in the generating function ${\left({\sum }_{i=1}^m{x}^i\right)}^{\ell }$, where $1⩽\ell ⩽n$. In this paper, we first show that the sequence of $(C_{k,1},C_{k,2},\dots ,C_{k,n})$ is unimodal. Then we use this as a tool to prove that the intersecting family in which every k-multiset contains a fixed element attains the maximum cardinality for $n⩾k+\lceil k/m\rceil$. In the special case when $m=1$ and $m=\infty$, our result gives rise to the famous Erdős-Ko-Rado Theorem and an unbounded multiset version for this problem given by Meagher and Purdy [11], respectively. The main result in this paper can be viewed as a bounded multiset version of the Erdős-Ko-Rado Theorem.