On the Running Time of Hypergraph Bootstrap Percolation

TL;DR

This paper investigates the maximum duration of hypergraph bootstrap percolation processes for complete hypergraphs, establishing bounds that depend on the size of the hypergraph and the uniformity parameter.

Contribution

It extends previous work from graphs to hypergraphs with uniformity $r \\geq 3$, providing bounds on the process's maximum running time for specific hypergraph sizes.

Findings

Maximum running time is (n^r) for certain hypergraph sizes.

Lower bounds are established for the case when hypergraph size is r+1.

Conjectures are made about the optimality of bounds for specific cases.

Abstract

Given $r\geq2$ and an $r$-uniform hypergraph $F$, the $F$-bootstrap process starts with an $r$-uniform hypergraph $H$ and, in each time step, every hyperedge which "completes" a copy of $F$ is added to $H$. The maximum running time of this process has been recently studied in the case that $r=2$ and $F$ is a complete graph by Bollob\'as, Przykucki, Riordan and Sahasrabudhe [Electron. J. Combin. 24(2) (2017), Paper No. 2.16], Matzke [arXiv:1510.06156v2] and Balogh, Kronenberg, Pokrovskiy and Szab\'o [arXiv:1907.04559v1]. We consider the case that $r\geq3$ and $F$ is the complete $r$-uniform hypergraph on $k$ vertices. Our main results are that the maximum running time is $\Theta\left(n^r\right)$ if $k\geq r+2$ and $\Omega\left(n^{r-1}\right)$ if $k=r+1$. For the case $k=r+1$, we conjecture that our lower bound is optimal up to a constant factor when $r=3$, but suspect that it can be improved by more than a constant factor for large $r$.