Some inequalities concerning cross-intersecting families of integer sequences

Let [n] = {1, 2, ... , n} and let [n]k be the family of integer sequences (x1, x2, ... , xk) with 1 & LE; xi & LE; n, i = 1, 2, ... , k. Two families F, Q & SUB; [n]k are called cross-intersecting if F and G agree in at least one position for every F & ISIN; F and G & ISIN; Q. A family F & SUB; [n]k is called nontrivial if for every i & ISIN; [k] there exist (x1, x2, ... , xk), (y1, y2, ... , yk) & ISIN; F such that xi = yi. In the present paper, we show that if F, Q & SUB; [n]k are non-empty cross-intersecting and n & GE; 2, then |F| + |Q| & LE;1 + nk- (n -1)k. If F, Q & SUB; [n]k are both non-trivial, cross-intersecting and n & GE; 2, then |F| + |Q| & LE; 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.& COPY; 2023 Elsevier B.V. All rights reserved.