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.

我们回答了 Gallai 在 1969 年提出的关于 Sperner 引理关键性的问题，该问题在 Jensen 和 Toft 的合集中列为问题 9.14（图形着色问题，Wiley，纽约，1995 年）。斯佩纳引理指出，如果带有标签\(1, 2, \ldots , d+1\)的d -单纯形\(\Delta ^d\)的三角剖分的顶点的标签具有以下属性： (i) 每个\(\Delta ^d\)的顶点接收一个不同的标签，并且 (ii) 位于\(\Delta ^d\)的面上的任何顶点与该面的顶点之一具有相同的标签，则存在彩虹面（顶点具有成对不同标签的面）。对于\(d\le 2\)，不难证明对于每个面\(\sigma \)，都存在具有上述属性的标签，其中\(\sigma \)是唯一的彩虹面。然而，对于每一个\(d\ge 3\)，我们构造了一个无限的例子族，但情况并非如此，这意味着加莱问题的答案作为推论。该构造基于 4 多胞体的性质，该性质早前曾被用来反驳莫茨金关于邻近多胞体的主张。