Poisson limit for strictly balanced subgraph counts ($X_H(G(n,c{n}^{-v/e}))\Rightarrow \mathrm{Po}(c^e/a)$)

Here $v,e,a$ are the numbers of vertices, edges and graph automorphisms of $H$. Vertex-disjoint ordered $r$-tuples contribute $(n{)}_{rv}{p}^{re}/a^r\to (c^e/a{)}^r$ to the factorial moment. Adding a copy overlapping the previous union in $f$ vertices and $g$ edges changes the power of $n$ by $-f+(v/e)g\le 0$, strictly when the intersection is a proper nonempty subgraph. The first overlap is strict because $H$ is connected and the previous copies are disjoint. Finitely many overlap types therefore contribute $o(1)$. The factorial-moment criterion for Poisson convergence gives the result.