We consider a generalization of Sperner's lemma for triangulations of m-discs whose vertices are colored in at most m colors. A coloring on the boundary (m-1)-sphere defines an element in the corresponding homotopy group of the sphere. Depending on this invariant, a lower bound is obtained for the number of fully colored simplexes. In particular, if the Hopf invariant is nonzero on the boundary of 4-disk, then there are at least 9 fully colored tetrahedra and if the Hopf invariant is d, then the lower bound is 3d + 3.
Musin, Oleg R. 2020. “Homotopy Groups and Quantitative Sperner-Type Lemma.” ArXiv:2007.08715 [Math], July. http://arxiv.org/abs/2007.08715.