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

Abstract

https://doi.org/10.48550/arXiv.2506.05581

Attending to an open problem in the literature stated by Mirzakhani and Vondr´ak, we give a lower bound of the number of nonmonochromatic 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.