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 $S_r^d$ of radius $r>1/\sqrt{2}$. The proof rests on the following statement, conjectured by F. Morić 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 $\Vert v_1-v_2\Vert >1$. The same holds for unit simplices on a $d$-dimensional sphere of radius greater than $1/\sqrt{2}$.

References in corpus (2)

• Unit distances and diameters in Euclidean spaces
• Diameter graphs in ${\mathbb{R}}^4$

Cited by in corpus (1)

• Some old and new problems in combinatorial geometry I: Around Borsuk's problem