# The maximum length of $K_r$-Bootstrap Percolation

## TL;DR

This paper investigates the maximum duration of the $K_r$-bootstrap percolation process on graphs, disproving a conjecture for all $r \\geq 6$ and providing improved bounds for $r=5$, using advanced combinatorial constructions.

## Contribution

The authors disprove the conjecture that the maximum running time is $o(n^2)$ for all $r$, and establish new lower bounds for $r=5$ and $r \\geq 6$ in bootstrap percolation.

## Key findings

- Disproved Bollobás et al.'s conjecture for all $r \\geq 6$.
- Provided improved lower bounds for $r=5$ using Behrend's construction.
- Showed that the maximum running time can be \\Omega(n^2) for certain $r$.

## Abstract

Graph-bootstrap percolation, also known as weak saturation, was introduced by Bollob\'as in 1968. In this process, we start with initial "infected" set of edges $E_0$, and we infect new edges according to a predetermined rule. Given a graph $H$ and a set of previously infected edges $E_t\subseteq E(K_n)$, we infect a non-infected edge $e$ if it completes a new copy of $H$ in $G=([n],E_t\cup e)$. A question raised by Bollob\'as asks for the maximum time the process can run before it stabilizes. Bollob\'as, Przykucki, Riordan, and Sahasrabudhe considered this problem for the most natural case where $H=K_r$. They answered the question for $r\leq 4$ and gave a non-trivial lower bound for every $r\geq 5$. They also conjectured that the maximal running time is $o(n^2)$ for every integer $r$. In this paper we disprove their conjecture for every $r\geq 6$ and we give a better lower bound for the case $r=5$; in the proof we use the Behrend construction.