Sub-Gaussian tails for the number of triangles in G(n,p)

Abstract

Let X be the random variable that counts the number of triangles in the random graph G(n,p). We show that for some absolute constant c, the probability that X deviates from its expectation by at least \lambda \var(X)^{1/2} is at most e^{-c\lambda^2}, provided that n^{-1}(\ln n)^{10} \le p \le n^{-1/2}(\ln n)^{-10}, \lambda = \omega(\ln n) and \lambda \le \min\{(np)^{1/2}, n^{-3/4}p^{-3/2}, n^{1/6}\}.