Aug 29, 2014
Sperner's lemma
Color the vertices of a triangulated triangle with three colors such that:
each vertex of the main triangle has a different color;
each vertex on an edge of the main triangle is colored with one of the two colors at the end of its edge;
then there exists a small triangle whose vertices are colored with all three different colors. More precisely, there exists an odd number of such triangles.
This result looks playful and innocent but is in fact quite powerful. It is known, for instance, to lead to an easy proof of Brouwer's fixed point theorem. Its power mainly lies in building bridges between discrete, combinatorial mathematics and continuous mathematics.
