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

Abstract

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 $\{1,2,\dots ,n\}$ such that $A_i\cap B_j=\varnothing$ and $|A_i\cap A_j|+|B_i\cap B_j|=k$ for all $i\ne j$. 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 finite projective plane.

7 pages, 2 figures