[2309.09973] Monochromatic boxes of unit volume
Abstract:Erdős 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.
COLORING AND DENSITY THEOREMS FOR CONFIGURATIONS OF A GIVEN VOLUME VJEKOSLAV KOVAČ arXiv:2309.09973v3 [math.CO] 14 Jan 2026 Abstract. This is a treatise on finite point configurations spanning a fixed volume to be found in a single color-class of an arbitrary finite (measurable) coloring of the Euclidean space…
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