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 G 0 , at each step we add to G 0 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) r + 1 with r >= 3, we provide a new construction for G 0 that shows that the number of steps of this process can be of order 9 (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 K (3) 4 minus an edge, then the maximum possible running time is 2n n - (sic)log 2 (n - 2)(sic) - 6. However, if H is K (3) 5 minus an edge, then the process can run for 9 (n 3) steps. (c) 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).