Homotopy groups and quantitative Sperner-type lemma

We prove quantitative forms of Sperner's lemma in which the boundary homotopy class controls the number of fully labeled simplices. Let a convex $(m+1)$-ball have a geometric triangulation labeled by $n+2$ colors, with $m\ge n\ge1$ and no fully labeled $(n+1)$-simplex on the boundary. The labeling i