all $0 \leq i \leq |E|$ and $j \leq i+1$;

3.  there is a finite integer $e_i^{(G,V,E)}$, independent of $n$, for which the exponent of $H_i(\operatorname{Conf}_n^{\operatorname{sink}}(G,V))$ is at most $e_i^{(G,V,E)}$.\

We note that the second part of Theorem \[sinkexpstab\] implies that the homological dimension of $\operatorname{Conf}_n^{\operatorname{sink}}(G,V)$ is equal to $|E|$ for $n \gg 0$. Indeed, Theorem \[fcm\] implies that $|E|$ is an upper bound on the homological dimension, while the second part of Theorem \[sinkexpstab\] implies that this bound is eventually realized. At this time it is unclear for what values of $n$ this homological dimension is realized, which is in contrast to the explicit bounds of Theorem \[dmtmainthm\].\
For the reader more knowledge about generalized representation stability in the sense of [@R2], it is interesting to note that if $G$ is a tree with vertex set $V$, then the homology groups $H_{|E|}(\operatorname{Conf}_{\bullet}^{\operatorname{sink}}(G,V))$ exhibit the full range of possible growth. That is to say, We may write $$\operatorname{rank}_{{\mathbb{Z}}}(H_i(\operatorname{Conf}_n^{\operatorname{sink}}(G,V)) = p_1^{(|E|)}(n) + p_2^{(|E|)}(n)2^n + \ldots + p_{|V|}^{(|E|)}(n)|V|^n,$$ where none of the polynomials $p_j^{(|E|)}$ are zero. To the knowledge of the author, sink configuration spaces are one of the first naturally arising examples of such behavior. Finally, note that computations done by 