Intersections and distinct intersections in cross-intersecting families

Let F, G be two cross-intersecting families of k-subsets of {1, 2, ... , n}. Let J (sic) G, I(J, G) denote the families of all intersections F boolean AND G with F is an element of F, G is an element of G, and all distinct intersections F boolean AND G with F not equal G, F is an element of F, G is an element of G, respectively. For a fixed T subset of {1, 2, ... , n}, let S-T be the family of all k-subsets of {1, 2, ... , n} containing T. In the present paper, we show that |F (sic) G | is maximized when F = G = S-{1} for n >= 2k(2)+8k, while surprisingly |I(F, G)| is maximized when F = S-{1,S-2} boolean OR S-{3,S-4} boolean OR S-{1,S-4,S-5} boolean OR S-{2,S-3,S-6} and G = S-{1,S-3} boolean OR S-{2,S-4} boolean OR S-{1,S-4,S-6} boolean OR S-{2,S-3,S-5} for n >= 100k(2). The maximum number of distinct intersections in a t-intersecting family is determined for n >= 3(t + 2)(3)k(2) as well. (c) 2022 Elsevier Ltd. All rights reserved.