= Poisson limit for triangles in a binomial random graph
{c}
{title2=$\binom n3p^3\to\alpha\Longrightarrow X\to\operatorname{Poisson}(\alpha)$}
Convergence holds in <total variation distance>. A triangle indicator depends only on its three edges, so its dependency neighborhood consists of itself and the $3(n-3)$ triangles sharing an edge. The <Poisson approximation with dependency neighborhoods> gives errors $b_1=O(n^4p^6)$ and $b_2=O(n^4p^5)$. Under the displayed scaling $p$ is of order $n^{-1}$, so both vanish. The probability of at least one triangle therefore tends to $1-e^{-\alpha}$.
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