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arXiv · 2404.04867

A Bollob\'as-type problem: from root systems to Erd\H{o}s-Ko-Rado

Patrick J. Browne·Qëndrim R. Gashi·Padraig Ó Catháin
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Abstract

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.

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Patrick J. Browne, Qëndrim R. Gashi, Padraig Ó Catháin. 2024-04-07. A Bollob\'as-type problem: from root systems to Erd\H{o}s-Ko-Rado. https://arxiv.org/abs/2404.04867

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