Sharp thresholds for contagious sets in random graphs

For fixed r >= 2, we consider bootstrap percolation with threshold r on the Erdos-Renyi graph G(n,p). We identify a threshold for p above which there is with high probability a set of size r that can infect the entire graph. This improves a result of Feige, Krivelevich and Reichman, which gives bounds for this threshold, up to multiplicative constants. As an application of our results, we obtain an upper bound for the threshold for K-4-percolation on G(n,p), as studied by Balogh, Bollobas and Morris. This bound is shown to be asymptotically sharp in subsequent work. These thresholds are closely related to the survival probabilities of certain time-varying branching processes, and we derive asymptotic formulae for these survival probabilities, which are of interest in their own right.