Maximal Fractional Cross-Intersecting Families

Given an irreducible fraction c d ∈ [ 0 , 1 ] , a pair ( A , B ) is called a c d -cross-intersecting pair of 2 [ n ] if A , B are two families of subsets of [ n ] such that for every pair A ∈ A and B ∈ B , | A ∩ B | = c d | B | . Mathew et al. (Graphs Comb 37:471–484, 2019) proved that | A | | B | ≤ 2 n if ( A , B ) is a c d -cross-intersecting pair of 2 [ n ] and characterized all the pairs ( A , B ) with | A | | B | = 2 n , such a pair also is called a maximal c d -cross-intersecting pair of 2 [ n ] , when c d ∈ { 0 , 1 2 , 1 } . In this note, we characterize all the maximal c d -cross-intersecting pairs ( A , B ) when 0 < c d < 1 and c d ≠ 1 2 , this result answers a question proposed by Mathew et al. (2019).