In this paper, oppositeness in spherical buildings is used to define an EKR-problem for flags in projective and polar spaces. A novel application of the theory of buildings and Iwahori-Hecke algebras is developed to prove sharp upper bounds for EKR-sets of flags. In this framework, we can reprove and generalize previous upper bounds for EKR-problems in projective and polar spaces. The bounds are obtained by the application of the Delsarte-Hoffman coclique bound to the opposition graph. The computation of its eigenvalues is due to earlier work by Andries Brouwer and an explicit algorithm is worked out. For the classical geometries, the execution of this algorithm boils down to elementary combinatorics. Connections to building theory, Iwahori-Hecke algebras, classical groups and diagram geometries are briefly discussed. Several open problems are posed throughout and at the end.

在本文中，球形建筑物中的对立性用于定义投影和极坐标空间中旗帜的 EKR 问题。开发了建筑物理论和 Iwahori-Hecke 代数的新应用来证明 EKR 标志集的明确上限。在这个框架中，我们可以在射影和极坐标空间中重新证明和推广 EKR 问题的先前上限。通过将 Delsarte-Hoffman 群应用绑定到对立图来获得边界。其特征值的计算归功于 Andries Brouwer 的早期工作，并制定了明确的算法。对于经典几何，该算法的执行归结为基本组合。简要讨论了与建筑理论、Iwahori-Hecke 代数、经典群和图表几何的联系。