  Let $G=(V,E)$ be a graph and $n$ a positive integer. Let $I_n(G)$ be the
abstract simplicial complex whose simplices are the subsets of $V$ that do not
contain an independent set of size $n$ in $G$. We study the collapsibility
numbers of the complexes $I_n(G)$ for various classes of graphs, focusing on
the class of graphs with maximum degree bounded by $\Delta$. As an application,
we obtain the following result:
  Let $G$ be a claw-free graph with maximum degree at most $\Delta$. Then,
every collection of
$\left\lfloor\left(\frac{\Delta}{2}+1\right)(n-1)\right\rfloor+1$ independent
sets in $G$ has a rainbow independent set of size $n$.
