The Deletion Method For Upper Tail Estimates

We present a new method to show concentration of the upper tail of random variables that can be written as sums of variables with plenty of independence. We compare our method with the martingale method by Kim and Vu, which often leads to similar results. Some applications are given to the number XG of copies of a graph G in the random graph G(n, p). In particular, for G = K4 and G = C4 we improve the earlier known upper bounds on − lnP(XK4 ≥ 2EXK4) in some range of p = p(n). .