A local central limit theorem for the number of triangles in a random   graph
  In this paper, we prove a local limit theorem for the distribution of the number of triangles in the Erdos-Renyi random graph $G(n,p)$, where $p \in (0,1)$ is a fixed constant. Our proof is based on bounding the characteristic function $\psi(t)$ of the number of triangles, and uses several different conditioning arguments for handling different ranges of $t$. 
Justin Gilmer, Swastik Kopparty
math.CO math.PR