Erdős-Ko-Rado theorem for bounded multisets

Abstract

Let k,m,n be positive integers with k⩾2. A k-multiset of [n]m is a collection of k integers from the set {1,2,…,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 (a1,a2,…,an) is said to be unimodal if there is some k∈{1,2,…,n}, such that a1⩽a2⩽…⩽ak−1⩽ak⩾ak+1⩾…⩾an. Given m,n,k, denote Ck,ℓ as the coefficient of xk in the generating function (∑i=1mxi)ℓ, where 1⩽ℓ⩽n. In this paper, we first show that the sequence of (Ck,1,Ck,2,…,Ck,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+⌈k/m⌉. In the special case when m=1 and m=∞, 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.