On Families of Weakly Cross-intersecting Set-pairs

Let F be a family of pairs of sets. We call it an (a, b)-set system if for every set-pair (A,B) in F we have that |A| = a, |B| = b, A ∩ B = ∅. The following classical result on families of cross-intersecting set-pairs is due to Bollobás [6]. Let F be an (a, b)-set system with the cross-intersecting property, i.e., for (Ai, Bi), (Aj, Bj) ∈ F with i 6= j we have that both Ai ∩ Bj and Aj ∩ Bi are non-empty. The maximum possible size of such a set system is ( a+b a ) , independent of the size of the ground set, and this bound is sharp. Surprisingly, the same upper bound holds even if we relax the cross-intersecting property, namely if for (Ai, Bi), (Aj, Bj) ∈ F we only require that Ai ∩ Bj 6= ∅ when i < j, as was shown in [7]. Several further generalizations were investigated in [2, 8, 10, 12]. For more details on the history and applications of this problem we refer to the surveys [14, 15] and Chapter 1 of [1]. In this paper we consider the following variant of the problem, introduced by Tuza [13].