Meunier Conjecture

TL;DR

This paper explores a generalization of Sperner's lemma proposed by Meunier, focusing on multicolored simplices and the existence of certain color combinations in subdivided simplices.

Contribution

It addresses Meunier's conjecture on multicolored Sperner lemmas, extending classical results to multiple colorings and analyzing the existence of specific color configurations.

Findings

Proves the existence of a simplex with specified color properties under certain conditions

Extends Sperner's lemma to multicolored and multi-configuration settings

Highlights open questions about connectivity of color hypergraphs

Abstract

Fr\'ed\'eric Meunier's question about a multicolored Sperner lemma is addressed, leaving the question of connectivity for the color hypergraphs of such a multicolored simplex. Sperner's lemma asserts the existence of a simplex using all the colors for any vertex coloring of a subdivision of a large simplex with appropriate boundary conditions. Meunier's questions generalizes this to the situation of having several such colorings and asserts the existence of a simplex using enough different colors from each coloring.