On Simplices in Diameter Graphs in R-4

A graph 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 R-4 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 R-4, 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.