A Tree Sperner Lemma

TL;DR

This paper introduces a combinatorial theorem for finite labellings of trees, linking it to fixed point theorems and KKM-type results, with applications in voting theory and extensions to infinite and cyclic structures.

Contribution

It establishes a new Tree Sperner Lemma and explores its equivalence to fixed point theorems and KKM-type theorems for trees and cycles, including social applications.

Findings

Proves a Tree Sperner Lemma for finite labellings.

Shows equivalence to fixed point theorems on metric trees.

Develops a KKM-type theorem for cycles and discusses social implications.

Abstract

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.