On Schur's Conjecture in $\mathbb R^4$

A diameter graph in $\mathbb R^d$ is a graph in which vertices are points of a finite subset of $\mathbb R^d$ and two vertices are joined by an edge if the distance between them is equal to the diameter of the vertex set. This paper is devoted to Schur's conjecture, which asserts that any diameter graph on $n$ vertices in $\mathbb R^d$ contains at most $n$ complete subgraphs of size $d$. It is known that Schur's conjecture is true in dimensions $d\le 3$. We prove this conjecture for $d=4$ and give a simple proof for $d=3$.