# Randomized construction of complexes with large diameter

## TL;DR

This paper introduces a probabilistic method to construct high-diameter simplicial complexes, significantly improving the lower bounds and advancing understanding of their combinatorial properties.

## Contribution

It provides a new probabilistic construction that yields near-optimal lower bounds on the diameter of simplicial complexes, improving previous bounds exponentially.

## Key findings

- New lower bound within O(d^2) factor of upper bound
- Improved bounds for pseudomanifolds
- Advances understanding of complex diameters

## Abstract

We consider the question of the largest possible combinatorial diameter among $(d-1)$-dimensional simplicial complexes on $n$ vertices, denoted $H_s(n, d)$. Using a probabilistic construction we give a new lower bound on $H_s(n, d)$ that is within an $O(d^2)$ factor of the upper bound. This improves on the previously best-known lower bound which was within a factor of $e^{\Theta(d)}$ of the upper bound. We also make a similar improvement in the case of pseudomanifolds.