  Motivated by an Erd\H{o}s--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
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.
