Some inequalities concerning cross-intersecting families of integer sequences

Abstract

Let [n]={1,2,…,n} and let [n]k be the family of integer sequences (x1,x2,…,xk) with 1≤xi≤n, i=1,2,…,k. Two families F,G⊂[n]k are called cross-intersecting if F and G agree in at least one position for every F∈F and G∈G. A family F⊂[n]k is called non-trivial if for every i∈[k] there exist (x1,x2,…,xk), (y1,y2,…,yk)∈F such that xi≠yi. In the present paper, we show that if F,G⊂[n]k are non-empty cross-intersecting and n≥2, then|F|+|G|≤1+nk−(n−1)k. If F,G⊂[n]k are both non-trivial, cross-intersecting and n≥2, then|F|+|G|≤nk−2(n−1)k+(n−2)k+2. We also establish a similar inequality for non-empty cross-intersecting families of generalized integer sequences.