Topologic proofs of some combinatorial theorems

AbstractDuring the last 50 years several combinatorial theorems have been proved which have provided elegant proofs of a number of fundamental results in topology. These include Sperner's Lemma, Tucker's Lemma, and Kuhn's Cubical Sperner Lemma, which have been applied to the Brouwer Fixed Point Theorem, as well as a number of results involving antipodal properties of continuous mappings.In this paper the reverse process is used to find topologic proofs of several combinatorial results. In response to several questions raised by Kuhn, a proof of Sperner's Lemma from the Brouwer Fixed Point Theorem is given, as is a proof of Tucker's Lemma from a topologic non-existence theorem for certain continuous mappings of an n-ball to its boundary. In the final section, a new labeling theorem for the n-cube, which is equivalent to Tucker's Lemma, is presented, and is proved by using topologic methods