Sperner's Lemma - Generalizations

Generalizations

Sperner's lemma has been generalized to colorings of polytopes with n vertices.. In that case, there are at least n-k fully labeled simplices, where k is the dimension of the polytope and "fully labeled" indicates that every label on the simplex has a different color. For example, if a polygon with n vertices is triangulated and colored according to the Sperner criterion, then there are at least n-2 fully labeled triangles. The general statement was conjectured by Atanassov in 1996, who proved it for the case k=2. The proof of the general case was first given by de Loera, Peterson, and Su in 2002.