The largest sets of non-opposite chambers in spherical buildings of type $B$
Jan De Beule, Philipp Heering, Sam Mattheus, Klaus Metsch
TL;DR
This paper addresses the problem of classifying largest families of non-opposite flags (EKR-sets) in finite spherical buildings of type $B$, recasting the center conjecture in an anti-design framework. Using a uniform antidesign-based approach combined with a graph-theoretic opposition framework and ratio bounds, it shows that for $\mathrm{PS}(n,e,q)$ with $e\ge 1$ or $n$ even and large $q$, the maximum EKR-sets of chambers are precisely blow-ups of maximum EKR-sets of points ($s=1$) or generators ($s=n$), with intermediate $s$-spaces not attaining the bound. The authors develop explicit antidesigns and transfer principles that connect chamber configurations to subspace configurations, enabling a complete classification in almost all polar spaces and highlighting exceptional open cases (notably $e=\tfrac{1}{2}$, $n$ odd). The work advances the algebraic approach to EKR problems in spherical buildings and provides a robust method with potential extensions to other types and ranks.
Abstract
The investigation into large families of non-opposite flags in finite spherical buildings has been a recent addition to a long line of research in extremal combinatorics, extending classical results in vector and polar spaces. This line of research falls under the umbrella of Erdős-Ko-Rado (EKR) problems, but poses some extra difficulty on the algebraic level compared to aforementioned classical results. From the building theory point of view, it can be seen as a variation of the center conjecture for spherical buildings due to Tits, where we replace the convexity assumption by a maximality condition. In previous work, general upper bounds on the size of families of non-opposite flags were obtained by applying eigenvalue and representation-theoretic techniques to the Iwahori-Hecke algebras of non-exceptional buildings. More recently, the classification of families reaching this upper bound in type $A_n$, for $n$ odd, was accomplished by Heering, Lansdown, and Metsch. For buildings of type $B$, the corresponding Iwahori-Hecke algebra is more complicated and depends non-trivially on the type and rank of the underlying polar space. Nevertheless, we are able to find a uniform method based on antidesigns and obtain classification results for chambers (i.e.\ maximal flags) in all cases, except type $^2A_{4n-3}$.