A Bollobás-type problem: from root systems to Erdős-Ko-Rado

Motivated by an Erdős--Ko--Rado type problem on sets of strongly orthogonal roots in the \(A_{\ell}\) root system, we estimate bounds for the size of a family of pairs \((A_{i}, B_{i})\) of \(k\)-subsets in \(\{ 1, 2, \ldots, n\}\) such that \(A_{i} \cap B_{j}= \emptyset\) and \(|A_{i} \cap A_{j}| + |B_{i} \cap B_{j}| = k\) for all \(i \neq j\). This is reminiscent of a classic problem of Bollobás. 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.