On the Running Time of Hypergraph Bootstrap Percolation

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