On simplices in diameter graphs in ℝ4

Abstract

Agraph G is a diameter graph in ℝ^d if its vertex set is a finite subset in ℝ^d of diameter 1 and edges join pairs of vertices a unit distance apart. It is shown that if a diameter graph G in ℝ⁴ contains the complete subgraph K on five vertices, then any triangle in G shares a vertex with K. The geometric interpretation of this statement is as follows. Given any regular unit simplex on five vertices and any regular unit triangle in ℝ⁴, then either the simplex and the triangle have a common vertex or the diameter of the union of their vertex sets is strictly greater than 1.