Moderate deviations of triangle counts in sparse Erdős-Rényi random graphs G(n, m) and G(n, p)

We consider the question of determining the probability of triangle count deviations in the Erdos-Renyi random graphs G (n,m) and G (n,p)with densities larger than n(-1/2)(log n)(1/2). In particular, we determine the log probability log P(N-Delta(G) > (1+delta)p(3)n(3))up to a constant factor across essentially the entire range of possible deviations, in both the G (n,m) and G (n,p) model. For the G(n,p) model, we also prove a stronger result, up to a(1+o(1)) factor, in the non-localised regime. We also obtain some results for the lower tail and for counts of cherries (paths of length 2).