An EKR-theorem for finite buildings of type $$D_{\ell }$$ D ℓ

Abstract

We define optimal EKR-sets in finite buildings. This definition is motivated by various contributions on optimal EKR-sets in finite projective spaces and polar spaces. Our main result is the classification of optimal EKR-sets of type $$\{ \ell , \ell -1\}$$ in finite building of type $$D_{\ell }$$ with $$\ell $$ even. As it is the case for most of the known optimal EKR-sets in finite buildings, our EKR-sets have a natural center. This provides some evidence that the EKR-problem for finite buildings and Tits center conjecture are closely related.