A Tree Sperner Lemma

In this paper we prove a combinatorial theorem for finite labellings of trees, and show that it is equivalent to a theorem for finite covers of metric trees and a fixed point theorem on metric trees. We trace how these connections mimic the equivalence of the Brouwer fixed point theorem with the classical KKM lemma and Sperner's lemma. We also draw connections to a KKM-type theorem about infinite covers of metric trees and fixed point theorems for non-compact metric trees. Finally, we develop a new KKM-type theorem for cycles, and discuss interesting social consequences, including an application in voting theory.