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=K_r+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(n^r). 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.