A local central limit theorem for the number of triangles in a random graph

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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 $t$.