A short proof of an Erdős–Ko–Rado theorem for compositions

Abstract

If a1,…,ak and n are positive integers such that n=a1+⋯+ak, then the tuple (a1,…,ak) is a composition ofn of lengthk. We say that (a1,…,ak)stronglyt-intersects(b1,…,bk) if there are at least t distinct indices i such that ai=bi. A set A of compositions is stronglyt-intersecting if every two members of A strongly t-intersect. Let Cn,k be the set of all compositions of n of length k. Ku and Wong (2013) showed that for every two positive integers k and t with k≥t+2, there exists an integer n0(k,t) such that for n≥n0(k,t), the size of any strongly t-intersecting subset A of Cn,k is at most n−t−1n−k, and the bound is attained if and only if A={(a1,…,ak)∈Cn,k:ai1=⋯=ait=1} for some distinct i1,…,it in {1,…,k}. We provide a short proof of this analogue of the Erdős–Ko–Rado Theorem. It yields an improved value of n0(k,t). We also show that the condition n≥n0(k,t) cannot be replaced by k≥k0(t) or n≥n0(t) (that is, the dependence of n on k is inevitable), and we suggest a Frankl-type conjecture about the extremal structures for any n, k, and t.