On simplices in diameter graphs in ℝ4
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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.
Keywordsdiameter graphs Schur’s conjecture
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