Complexes of nearly maximum diameter

TL;DR

This paper demonstrates the existence of high-diameter $d$-dimensional simplicial complexes, nearly reaching the theoretical maximum, using probabilistic methods, and also determines the asymptotic maximum diameter for $d$-pseudomanifolds.

Contribution

It provides a probabilistic proof of complexes with near-maximum diameter and establishes the first-order asymptotics for pseudomanifolds.

Findings

Existence of $d$-complexes with diameter close to the upper bound.

Maximum diameter of $d$-pseudomanifolds on $n$ vertices determined.

Diameter bounds are tight up to the first order.

Abstract

The diameter of a strongly connected $d$-dimensional simplicial complex is the diameter of its dual graph. We provide a probabilistic proof of the existence of $d$-dimensional simplicial complexes with diameter $ (\frac{1}{d \cdot d!} - (\log n)^{-\epsilon}) n^d$. Up to the first order term, this is the best possible lower bound for the maximum diameter of a $d$-complex on $n$ vertices as a simple volume argument shows that the diameter of a $d$-dimensional simplicial complex is at most $ \frac{1}{d} \binom{n}{d}$. We also find the right first-order asymptotics for the maximum diameter of a $d$-pseudomanifold on $n$ vertices.