On the Upper Tail of Counts of Strictly Balanced Subgraphs

Let XG be the number of copies of G in the Erdős-Rényi binomial random graph G(n, p). Janson, Oleszkiewicz and Ruciński proved that for every t > 1 exp{−Ot(M G ln(1/p))} 6 P {XG > tEXG} 6 exp{−Ωt(M G)}, where M∗ G is a certain function of n and p. For G = K3 the logarithmic gap between the bounds was closed by Chatterjee and, independently, DeMarco and Kahn. We provide matching bounds for strictly balanced G, when EXG 6 lnn. Also, we give matching bounds for C4, K4, and stars K1,k in a broader range of EXG. In particular, this improves some results of Janson and Ruciński for which the so called deletion method was used.