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 $A_{\mathrm{\ell }}$ root system, we estimate bounds for the size of a family of pairs $(A_i,B_i)$ of $k$-subsets in $\{1,2,\dots ,n\}$ such that $A_i\cap B_j=\mathrm{\varnothing }$ and $\mathrm{\mid }{A}_i\cap A_j\mathrm{\mid }+\mathrm{\mid }{B}_i\cap B_j\mathrm{\mid }=k$ for all $i\mathrm{\ne }j$. 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.