On the Running Time of Hypergraph Bootstrap Percolation

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