Diameter bound for facet-ridge incidence graphs of geometric lattices

This paper proves that the facet-ridge incidence graph of the order complex of any finite geometric lattice of rank $r$ has diameter at most $\left(\genfrac{}{}{0px}{}{r}{2}\right)$. A key ingredient is the well-known fact that every ordering of the atoms of any finite geometric lattice gives rise to a lexicographic shelling of its order complex. The paper also gives results that provide some evidence that this bound ought to be sharp as well as examples indicating that the question of sharpness is quite subtle.