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

TL;DR

This paper establishes a lower bound on the number of non-monochromatic simplices in Sperner labelings of a specific triangulation of a k-simplex, addressing an open problem in combinatorial topology.

Contribution

It provides a new lower bound for Sperner labelings on a maximally triangulated k-simplex with integer vertices, solving an open problem.

Findings

Lower bound on non-monochromatic simplices established

Addresses an open problem in the literature

Focuses on triangulations with maximum simplices

Abstract

Attending to an open problem in the literature stated by Mirzakhani and Vondr\'ak, 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.