Erd\H{o}s and Graham asked whether, for any coloring of the Euclidean plane $\mathbb{R}^2$ in finitely many colors, some color class contains the vertices of a rectangle of every given area. We give the negative answer to this question and its higher-dimensional generalization: there exists a finite coloring of the Euclidean space $\mathbb{R}^n$, $n\geq2$, such that no color class contains the $2^n$ vertices of a rectangular box of volume $1$. The present note is a very preliminary version of a longer treatise on similar problems.
