Fair splittings by independent sets in sparse graphs

Given a partition V-1 square cup V-2 square cup ... square cup 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. Further, 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.