Sharp results concerning disjoint cross-intersecting families

Abstract

For an n-element set X let Xk be the collection of all its k-subsets. Two families of sets A and B are called cross-intersecting if A∩B≠0̸ holds for all A∈A, B∈B. Let f(n,k) denote the maximum of min{|A|,|B|} where the maximum is taken over all pairs of disjoint, cross-intersecting families A,B⊂[n]k. Let c=log2e. We prove that f(n,k)=12n−1k−1 essentially iff n>ck2 (cf. Theorem 1.4 for the exact statement). Let f∗(n,k) denote the same maximum under the additional restriction that the intersection of all members of both A and B are empty. For k≥6 and n≥k3−k2 we show that f∗(n,k)=12n−1k−1−n−2kk−1+1 and the restriction on n is essentially sharp (cf. Theorem 1.5).