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