Long running times for hypergraph bootstrap percolation

TL;DR

This paper investigates the maximum duration of hypergraph bootstrap percolation processes, providing constructions and bounds for different hypergraph configurations, revealing that process duration can vary significantly.

Contribution

It introduces new constructions and bounds for the number of steps in hypergraph bootstrap percolation, answering open questions about process duration for specific hypergraphs.

Findings

For certain hypergraphs, the process can take on the order of n^r steps.

Maximum steps for some hypergraphs are tightly bounded, e.g., 2n - log_2(n) for specific cases.

Different hypergraph structures lead to vastly different process durations.

Abstract

Consider the hypergraph bootstrap percolation process in which, given a fixed $r$-uniform hypergraph $H$ and starting with a given hypergraph $G_0$, at each step we add to $G_0$ all edges that create a new copy of $H$. We are interested in maximising the number of steps that this process takes before it stabilises. For the case where $H=K_{r+1}^{(r)}$ with $r\geq3$, we provide a new construction for $G_0$ that shows that the number of steps of this process can be of order $\Theta(n^r)$. This answers a recent question of Noel and Ranganathan. To demonstrate that different running times can occur, we also prove that, if $H$ is $K_4^{(3)}$ minus an edge, then the maximum possible running time is $2n-\lfloor \log_2(n-2)\rfloor-6$. However, if $H$ is $K_5^{(3)}$ minus an edge, then the process can run for $\Theta(n^3)$ steps.