Abstract
Given a partition V1 ⊔ V2 ⊔... ⊔Vm 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 Vj. 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 Vj in certain sparse graphs for q a power of a prime.
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