On a problem of Specker about Euclidean representations of finite graphs

Say that a graph $G$ is representable in ${\mathbb{R}}^n$ if there is a map $f$ from its vertex set into the Euclidean space ${\mathbb{R}}^n$ such that $6f\left(x\right)?f\left(x^{\prime }\right)6=6f\left(y\right)?f\left(y^{\prime }\right)6$ iff $\{x,x^{\prime }\}$ and$\{y,y^{\prime }\}$ are both edges or both non-edges in $G$. The purpose of this note is to present the proof of the following result, due to Einhorn and Schoenberg in Einhorn and Schoenberg (1966): if $G$ finite is neither complete nor independent, then it is representable in ${\mathbb{R}}^{\left|G\right|?2}$. A similar result also holds in the case of finite complete edge-colored graphs.