Poisson limit for triangles in a binomial random graph ($\left(\genfrac{}{}{0ex}{}{n}{3}\right)p^3\to \alpha ⟹X\to \mathrm{Poisson}(\alpha )$)

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 $b_1=O(n^4{p}^6)$ and $b_2=O(n^4{p}^5)$. Under the displayed scaling $p$ is of order $n^{-1}$, so both vanish. The probability of at least one triangle therefore tends to $1-e^{-\alpha }$.