Slow graph bootstrap percolation III: Chain constructions

For graphs $H$, we study the extremal function $M_H(n)$ which is the maximum running time (until stabilisation) of an $H$-bootstrap percolation process on $n$ vertices. Building on previous work in the clique case $H=K_k$, we develop a general framework of chain constructions. We demonstrate the flexibility of this framework by applying several variations of the method to give lower bounds on $M_H(n)$ for a wide variety of different graphs $H$ including dense graphs, random graphs and complete bipartite graphs. In particular, we focus on the question of whether $M_H(n)$ is (almost) quadratic or not and our lower bounds develop connections with additive combinatorics, utilising constructions of sets free of solutions to certain linear equations. Finally, our lower bounds are complemented by upper bounds which connect $M_H(n)$ to other problems in extremal graph theory such as the Ruzsa-Szemer\'edi (6,3)-Theorem.