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

If a 1 , … , a k and n are positive integers such that n = a 1 + ⋯ + a k , then the tuple ( a 1 , … , a k ) is a composition of n of length k . We say that ( a 1 , … , a k ) strongly t -intersects ( b 1 , … , b k ) if there are at least t distinct indices i such that a i = b i . A set A of compositions is strongly t -intersecting if every two members of A strongly t -intersect. Let C n , 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 n 0 ( k , t ) such that for n ≥ n 0 ( k , t ) , the size of any strongly t -intersecting subset A of C n , k is at most n − t − 1 n − k , and the bound is attained if and only if A = { ( a 1 , … , a k ) ∈ C n , k : a i 1 = ⋯ = a i t = 1 } for some distinct i 1 , … , i t in { 1 , … , k } . We provide a short proof of this analogue of the Erdős–Ko–Rado Theorem. It yields an improved value of n 0 ( k , t ) . We also show that the condition n ≥ n 0 ( k , t ) cannot be replaced by k ≥ k 0 ( t ) or n ≥ n 0 ( 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 .