A simple proof of almost percolation on G(n;p)

We consider bootstrap percolation on the binomial random graph $G(n,p)$ with infection threshold $rin mathbb{N}$, an infection process which starts from a set of initially infected vertices and in each step every vertex with at least $r$ infected neighbours becomes infected. We improve the results of Janson, L uczak, Turova, and Valier (2012) by strengthening the probability bounds on the number of infected vertices at the end of the process, using simple arguments based on martingales and giant components.