On the minimum number of non-monochromatic simplices for Sperner labelings of a regular triangulation

Abstract

Attending to an open problem in the literature stated by Mirzakhani and Vondrák, we give a lower bound of the number of non-monochromatic simplices for Sperner labelings of the vertices of a triangulation of a given $k$-simplex with vertices of integer coordinates. This triangulation maximizes the number of simplices over all the triangulations of the $k$-simplex with vertices of integer coordinates.