Problem #WSP-5333

Problem

Sperner’s lemma in dimension $2$.
Subdivide a triangle $ABC$ arbitrarily into a triangulation consisting of smaller triangles meeting edge to edge. Define Sperner coloring of the triangulation as an assignment of three colors to the vertices of the triangulation such that:

• Each of the three vertices $A,B,$ and $C$ of the initial triangle has a distinct color

• The vertices that lie along any edge of triangle $ABC$ have only two colors, the two colors at the endpoints of the edge. For example, each vertex on $AC$ must have the same color as $A$ or $C.$

Here is an example of Sperner’s triangulation

Prove that every Sperner coloring of every triangulation has at least one "rainbow triangle", a smaller triangle in the triangulation that has its vertices colored with all three different colors. More precisely, there must be an odd number of rainbow triangles.