Some Combinatorial Lemmas in Topology

For many years it has been known that a combinatorial result, called the Sperner Lemma, provides an elegant proof of the Brouwer Fixed Point Theorem. Although the proof is elementary, its complete formal exposition depends upon the somewhat complicated operation of subdividing a simplex. Also, the proof does not show whether the Sperner Lemma can be derived from the Brouwer Fixed Point Theorem. This central result of this paper is a combinatorial proposition, analogous to the Sperner Lemma, and applying to the n-cube, for which subdivision is a trivial operation. This Cubical Sperner Lemma follows immediately from the Brouwer Fixed Point Theorem and thus opens the possibility of other applications of topology to combinatorial problems. The question of such a topological proof is raised for another cubical analogue of the Sperner Lemma, due to Ky Fan, and for the Tucker Lemma, which is related to the antipodal point theorems. The Cubical Sperner Lemma of this paper implies the Tucker Lemma in 2-dimensions; this suggests that other connections joining these combinatorial results remain to be discovered.