Linear algebra and bootstrap percolation

In H -bootstrap percolation, a set A ⊂ V ( H ) of initially ‘infected’ vertices spreads by infecting vertices which are the only uninfected vertex in an edge of the hypergraph H . A particular case of this is the H-bootstrap process, in which H encodes copies of H in a graph G. We find the minimum size of a set A that leads to complete infection when G and H are powers of complete graphs and H encodes induced copies of H in G. The proof uses linear algebra, a technique that is new in bootstrap percolation, although standard in the study of weakly saturated graphs, which are equivalent to (edge) H-bootstrap percolation on a complete graph.