Non-trivial t-intersecting separated families

Let n, k, l, t be positive integers with k >= l >= t+2 and let X = X-1 & uplus;X-2 & uplus; <middle dot><middle dot><middle dot> & uplus;X-k, |X-i| = n. A family F of l-subsets of X is called a separated family if |F boolean AND X-i| <= 1 for all F is an element of F and i = 1, 2, ... , k. A separated family F is called non-trivial t-intersecting if |F boolean AND F '| >= t for all F, F ' is an element of F and | boolean AND {F : F is an element of F}| < t. In the present paper, we determine the maximum size of a non-trivial t-intersecting separated family for n > (t+1)(l-t-1)/2(k-t-1) + 1.(c) 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).