Geodesic
On Simplices in Diameter Graphs in~$\mathbb R^4$
Matematičeskie zametki, Tome 101 (2017) no. 2, pp. 232-246

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A graph $G$ is a diameter graph in $\mathbb R^d$ if its vertex set is a finite subset in $\mathbb R^d$ of diameter $1$ and edges join pairs of vertices a unit distance apart. It is shown that if a diameter graph $G$ in $\mathbb R^4$ 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 $\mathbb R^4$, 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 $1$.
Keywords: diameter graphs, Schur's conjecture. 
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     title = {On {Simplices} in {Diameter} {Graphs} in~$\mathbb R^4$},
     journal = {Matemati\v{c}eskie zametki},
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A. B. Kupavskii; A. A. Poljanskij. On Simplices in Diameter Graphs in~$\mathbb R^4$. Matematičeskie zametki, Tome 101 (2017) no. 2, pp. 232-246. http://geodesic.mathdoc.fr/item/MZM_2017_101_2_a8/