= Bounded-degree pairing avoidance estimate
{title2=$\Pr(\text{simple and avoiding }G_0)=\exp(-\lambda/2-\lambda^2/4-\mu)+o(1)$}
For bounded <vertex degrees>, total degree $M\to\infty$, and a forbidden graph $G_0$ of bounded <maximum degree>, put $\lambda=M^{-1}\sum_i d_i(d_i-1)$ and $\mu=M^{-1}\sum_{ij\in E(G_0)}d_i d_j$. Count prescribed loop pairs, double-edge pairs, and forbidden-edge pairs in the <configuration model>. Their means are respectively $\lambda/2+o(1)$, $\lambda^2/4+o(1)$, and $\mu+o(1)$. A fixed collection of $t$ disjoint prescribed pairs occurs with probability $[(M-1)(M-3)\cdots(M-2t+1)]^{-1}$. Bounded degrees make intersecting bad configurations contribute $O(M^{-1})$ to each fixed <factorial moment>. Thus the total bad-pattern count has factorial moments $(\lambda/2+\lambda^2/4+\mu)^j+o(1)$. The <Bonferroni inequalities> and uniformly bounded means give the displayed zero-pattern probability, without requiring the parameters to converge.
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