Proof of Schur's conjecture in ${\mathbb{R}}^d$

In this paper we prove Schur’s conjecture in ${\mathbb{R}}^d$, which states that any diameter graph $G$ in the Euclidean space ${\mathbb{R}}^d$ on $n$ vertices may have at most $n$ cliques of size $d$. We obtain an analogous statement for diameter graphs with unit edge length on a sphere $S_r^d$ of radius $r>1/\sqrt 2$. The proof rests on the following statement, conjectured by F. Mori'c and J. Pach: given two unit regular simplices ${\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2$ on $d$ vertices in ${\mathbb{R}}^d$, either they share $d-2$ vertices, or there are vertices $v_1\in {\mathrm{\Delta }}_1,v_2\in {\mathrm{\Delta }}_2$ such that $\mathrm{\parallel }{v}_1-v_2\mathrm{\parallel }>1$. The same holds for unit simplices on a $d$-dimensional sphere of radius greater than $1\mathrm{/}\sqrt{2}$.