Long running times for hypergraph bootstrap percolation

Consider the hypergraph bootstrap percolation process in which, given a fixed r-uniform hypergraph H and starting with a given hypergraph G0, at each step we add to G0 all edges that create a new copy of H. We are interested in maximising the number of steps that this process takes before it stabilises. For the case where H=Kr+1(r) with r≥3, we provide a new construction for G0 that shows that the number of steps of this process can be of order Θ(nr). This answers a recent question of Noel and Ranganathan. To demonstrate that different running times can occur, we also prove that, if H is K4(3) minus an edge, then the maximum possible running time is 2n−⌊log2(n−2)⌋−6. However, if H is K5(3) minus an edge, then the process can run for Θ(n3) steps.