Matej Stehlik

Sperner’s lemma states that in any labelling of the vertices of a triangulation of a d-simplex with d+1 labels, such that each vertex of the d-simplex receives a distinct label and any vertex lying in a face of the d-simplex has the same label as one of the vertices of that face, there exists a rainbow facet (a facet whose vertices have pairwise distinct labels). Tibor Gallai proved (in a different but equivalent form) that for d=1 and d=2, we can pick any facet of the triangulation and find a labelling where that facet is the unique rainbow simplex. In this talk, I will show that this is not the case for higher dimensions, thereby giving a negative answer to a question of Gallai from 1969. The construction is based on the properties of a 4-polytope which had been used earlier to disprove a claim of Theodore Motzkin on neighbourly polytopes.

 

Joint work with Tomas Kaiser and Riste Skrekovski.