Abstract
We answer a question posed by Gallai in 1969 concerning criticality in Sperner’s lemma, listed as Problem 9.14 in the collection of Jensen and Toft (Graph coloring problems, Wiley, 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 \(d\le 2\), 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 \(d\ge 3\), 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 Motzkin on neighbourly polytopes.
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Acknowledgements
We are indebted to Frank Lutz for the valuable information on small triangulations of manifolds collected in [15]. We used Sage to test these triangulations (as well as those obtained from [2, 11]) for specific properties, and we thank the developers of this fine piece of software. Finally, we would like to thank the referees for making some useful suggestions to improve the readability of the paper.
Funding
T. Kaiser: Supported by project GA20-09525 S of the Czech Science Foundation. M. Stehlík: Partially supported by ANR project DISTANCIA (ANR-17-CE40-0015). R. Škrekovski: Partially supported by the Slovenian Research Agency Program P1-0383, Project J1-3002, and BI-FR/22-23-Proteus-011
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To the memory of Frank H. Lutz (1968–2023).
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Kaiser, T., Stehlík, M. & Škrekovski, R. Criticality in Sperner’s Lemma. Combinatorica (2024). https://doi.org/10.1007/s00493-024-00104-4
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DOI: https://doi.org/10.1007/s00493-024-00104-4