Criticality in Sperner's Lemma

TL;DR

This paper investigates the criticality in Sperner's lemma by constructing examples that challenge previous assumptions, providing new insights into the lemma's behavior in higher dimensions.

Contribution

It answers Gallai's 1969 question by constructing infinite examples for dimensions three and above, demonstrating the non-uniqueness of rainbow facets.

Findings

For dimensions d ≥ 3, there exist infinite examples with multiple rainbow facets.

The construction uses properties of a specific 4-polytope related to neighborly polytopes.

The results extend understanding of Sperner's lemma beyond low-dimensional cases.

Abstract

We answer a question posed by T. Gallai in 1969 concerning criticality in Sperner's lemma, listed as Problem 9.14 in the collection of Jensen and Toft [Graph coloring problems, John Wiley & Sons, Inc., New York, 1995]. Sperner's lemma states that if a labelling of the vertices of a triangulation of the $d$-simplex $\Delta^d$ with labels $1, 2, \ldots, d+1$ has the property that (i) each vertex of $\Delta^d$ receives a distinct label, and (ii) any vertex lying in a face of $\Delta^d$ has the same label as one of the vertices of that face, then there exists a rainbow facet (a facet whose vertices have pairwise distinct labels). For $d\leq 2$, it is not difficult to show that for every facet $\sigma$, there exists a labelling with the above properties where $\sigma$ is the unique rainbow facet. For every $d\geq 3$, however, we construct an infinite family of examples where this is not the case, which implies the answer to Gallai's question as a corollary. The construction is based on the properties of a $4$-polytope which had been used earlier to disprove a claim of T. S. Motzkin on neighbourly polytopes.