The Maximum Diameter of Pure Simplicial Complexes and Pseudo-manifolds

Francisco Criado, Francisco Santos

Abstract: We construct d-dimensional pure simplicial complexes and pseudo-manifolds (without boundary) with n vertices whose combinatorial diameter grows as \(c_d n^{d-1}\) for a constant \(c_d\) depending only on d, which is the maximum possible growth. Moreover, the constant \(c_d\) is optimal modulo a singly exponential factor in d. The pure simplicial complexes improve on a construction of the second author that achieved \(c_d n^{2d/3}\). For pseudo-manifolds without boundary, as far as we know, no construction with diameter greater than \(n^2\) was previously known.