Mathematics – Combinatorics
Scientific paper
2009-09-02
Mathematics
Combinatorics
18 pages, 4 figures. See also related work at http://www.math.hmc.edu/~su/papers.html
Scientific paper
In this paper we prove a new combinatorial theorem for labellings of trees, and show that it is equivalent to a KKM-type theorem for finite covers of trees and to discrete and continuous fixed point theorems on finite trees. This is in analogy with the equivalence of the classical Sperner's lemma, KKM lemma, and the Brouwer fixed point theorem on simplices. Furthermore, we use these ideas to develop new KKM and fixed point theorems for infinite covers and infinite trees. Finally, we extend the KKM theorem on trees to an entirely new KKM theorem for cycles, and discuss interesting social consequences, including an application in voting theory.
Niedermaier Andrew
Rizzolo Douglas
Su Francis Edward
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