A Bollob\'as-type problem: from root systems to Erd{o}s-Ko-Rado

Motivated by an Erd{o}s--Ko--Rado type problem on sets of strongly orthogonal roots in the A_ root system, we estimate bounds for the size of a family of pairs (Aᵢ, Bᵢ) of k-subsets in \ 1, 2, …, n\ such that Aᵢ ∩ Bⱼ= ∅ and |Aᵢ ∩ Aⱼ| + |Bᵢ ∩ Bⱼ| = k for all i ≠ j. This is reminiscent of a classic problem of Bollob\'as. We provide upper and lower bounds for this problem, relying on classical results of extremal combinatorics and an explicit construction using the incidence matrix of a finite projective plane.