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Proof of Schur's conjecture in $\mathbb R^d$ (1402.3694v4)

Published 15 Feb 2014 in math.MG, cs.DM, and math.CO

Abstract: In this paper we prove Schur's conjecture in $\mathbb Rd$, which states that any diameter graph $G$ in the Euclidean space $\mathbb Rd$ 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 $Sd_r$ 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 $\mathbb Rd$, 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 $1/\sqrt 2$.

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