Upper bounds on the running time of bootstrap percolation

TL;DR

This paper establishes new upper bounds on the maximum running time of bootstrap percolation processes in hypergraphs, linking it to Turán densities and improving upon trivial bounds for complete graphs.

Contribution

It provides the first non-trivial upper bounds for the maximum running time of bootstrap percolation, connecting it to Turán densities for general hypergraphs.

Findings

For $t ext{-graphs}$, $M_{K_t}(n) o (rac{t-3}{t-2})inom{n}{2}$ as $n$ grows.

General bound: $M_F(n) o ig( ext{Turán density of }F-eig)inom{n}{k}$.

Improves the trivial bound $inom{n}{2}$ for $t ext{-graphs}$ with $t extgreater 4$.

Abstract

For $k$-graphs $F$ and $H_0$ the $F$-bootstrap percolation process (or $F$-process) starting with $H_0$ is a sequence $(H_i)_{i\geq0}$ of $k$-graphs such that $H_{i+1}$ is obtained from $H_i$ by adding all those $e\in V(H_0)^{(k)}\setminus E(H_i)$ as edges that complete a new copy of $F$. The running time of this $F$-process, denoted by $M_F(H_0)$, is the smallest $i$ with $H_i=H_{i+1}$. Bollob\'as proposed the problem of determining the maximum running time for $n\in\mathbb{N}$, i.e., $M_F(n)=\max_{\vert V(H_0)\vert=n}M_F(H_0)$. Although this problem has received a lot of attention recently, until now the best known upper bound for $M_{K_t}(n)$, with $t\geq5$, was the trivial bound $\binom{n}{2}$. Here we provide the first non-trivial upper bound for this problem by showing that $$M_{K_t}(n)\leq\Big(\frac{t-3}{t-2}+o(1)\Big)\binom{n}{2}$$ holds for every integer $t\geq 3$. In fact, we prove the following more general result. For every $k\geq2$, every $k$-graph $F$, and every $e\in E(F)$ we have $M_F(n)\leq\big(\pi(F-e)+o(1)\big)\binom{n}{k}$, where $\pi$ is the Tur\'an density.