Sperner's Lemma - Two-dimensional Case

Two-dimensional Case

The two-dimensional case is the one referred to most frequently. It is stated as follows:

Given a triangle ABC, and a triangulation T of the triangle. The set S of vertices of T is colored with three colors in such a way that

1. A, B and C are colored 1, 2 and 3 respectively
2. Each vertex on an edge of ABC is to be colored only with one of the two colors of the ends of its edge. For example, each vertex on AC must have a color either 1 or 3.

Then there exists a triangle from T, whose vertices are colored with the three different colors. More precisely, there must be an odd number of such triangles.