On the Sperner lemma

On the Complexity of the Sperner Lemma

On the Sperner Lemma and Its Applications

This paper presents elementary combinatorial proofs of Sperner’s Lemma and discusses several non-trivial theorems to which it has been applied. Examples are the Brouwer Fixed Point Theorem, the Fundamental Theorem of Algebra, and the solution to the Cake-cutting problem.

A Sperner lemma complete for PPA

The class PPA characterizes search problems whose solution is guaranteed by the lemma that \every nite graph with an odd degree vertex has another." The smaller class PPAD is de ned similarly for directed graphs. While PPAD has several natural complete problems corresponding to classical existence theorems in topology, no such complete problems were known for PPA. Here we overcome the di culty ...

Extensions of Sperner and Tucker's lemma for manifolds

The Sperner and Tucker lemmas are combinatorial analogous of the Brouwer and Borsuk Ulam theorems with many useful applications. These classic lemmas are concerning labellings of triangulated discs and spheres. In this paper we show that discs and spheres can be substituted by large classes of manifolds with or without boundary.

Games on the Sperner Triangle

We create a new two-player game on the Sperner Triangle based on Sperner’s lemma. Our game has simple rules and several desirable properties. First, the game is always certain to have a winner. Second, like many other interesting games such as Hex and Geography, we prove that deciding whether one can win our game is a PSPACE-complete problem. Third, there is an elegant balance in the game such ...