  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}\}.
