Upper Tail Bounds for Stars

For r >= 2, let X be the number of r-armed stars K-1,K-r in the binomial random graph G(n,p). We study the upper tail P(X >= (1 + epsilon)EX), and establish exponential bounds which are best possible up to constant factors in the exponent (for the special case of stars K-1,K-r this solves a problem of Janson and Rucinski, and confirms a conjecture by DeMarco and Kahn). In contrast to the widely accepted standard for the upper tail problem, we do not restrict our attention to constant epsilon, but also allow for epsilon >= n(-alpha) deviations.