A refined result on cross-intersecting families

Let A and B be two families of sets. A is t-intersecting if |A1 & AND; A2| & GE; t for all A1, A2 & ISIN; A. A and B are cross-intersecting if A & AND; B= null for all A & ISIN; A and B & ISIN; B. Let n, k, t & GE; 1 be non-negative integers with n & GE; 2k + t - 1. Let A & SUBE; ( [n] ) and B & SUBE; ([n]). Frankl proved that if A is t-intersecting and |A| & GE; 1, then |A| + |B| & LE; (n k ) + 1. In this paper, we sharpen Frankl's inequality when |A| & GE; 2, that is, if A is t-intersecting and |A| & GE; 2, then |A| + |B| & LE; (n)- (n-k-t+1) - (n-k-t) + 2 for k & GE; 3. Furthermore, we also characterize all extremal families achieving the extremal bounds.& COPY; 2023 Elsevier B.V. All rights reserved.