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

Abstract

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 $S^d_r$ 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 $\Delta_1,\Delta_2$ on $d$ vertices in $\mathbb R^d$, either they share $d-2$ vertices, or there are vertices $v_1\in \Delta_1,v_2\in \Delta_2$ such that $\|v_1-v_2\|>1$. The same holds for unit simplices on a $d$-dimensional sphere of radius greater than $1/\sqrt 2$.