  Given a partition $V_1 \sqcup V_2 \sqcup \dots \sqcup V_m$ of the vertex set
of a graph, we are interested in finding multiple disjoint independent sets
that contain the correct fraction of vertices of each $V_j$. We give conditions
for the existence of $q$ such independent sets in terms of the topology of the
independence complex. We relate this question to the existence of $q$-fold
points of coincidence for any continuous map from the independence complex to
Euclidean space of a certain dimension, and to the existence of equivariant
maps from the $q$-fold deleted join of the independence complex to a certain
representation sphere of the symmetric group. As a corollary we derive the
existence of $q$ pairwise disjoint independent sets accurately representing the
$V_j$ in certain sparse graphs for $q$ a power of a prime.
