Criticality in Sperner’s Lemma

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.