The maximum diameter of pure simplicial complexes and pseudo-manifolds

TL;DR

This paper constructs high-dimensional pure simplicial complexes and pseudo-manifolds with maximal combinatorial diameter growth proportional to $n^{d-1}$, establishing optimal bounds and improving previous results.

Contribution

It provides the first constructions achieving maximal diameter growth of order $n^{d-1}$ for pure complexes and pseudo-manifolds, with optimal constants.

Findings

Constructed complexes with diameter $c_d n^{d-1}$

Improved previous diameter bounds for pure complexes

Established new upper bounds for pseudo-manifolds without boundary

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.