Coloring and density theorems for configurations of a given volume

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 $\mathbb{R}^n$, or in a single large measurable subset $A\subseteq\mathbb{R}^n$. More specifically, we study vertex-sets of simplices, rectangular boxes, and parallelotopes, attempting to make progress on several open problems posed in the 1970s and the 1980s. As one of the highlights, we give the negative answer to a question of Erd\H{o}s and Graham, by coloring the Euclidean plane $\mathbb{R}^2$ in $25$ colors without creating monochromatic rectangles of unit area. More generally, we construct a finite coloring of the Euclidean space $\mathbb{R}^n$ such that no color-class contains the $2^m$ vertices of any (possibly rotated) $m$-dimensional rectangular box of volume $1$. A positive result is still possible if rectangular boxes of merely sufficiently large volumes are sought in a single color-class of a finite measurable coloring of $\mathbb{R}^n$, and we establish it under an additional assumption $n\geq m+1$. Also, motivated by a question of Graham on reasonable bounds in his result on monochromatic axes-aligned right-angled $m$-dimensional simplices, we establish its measurable coloring and density variants with polylogarithmic bounds, again in dimensions $n\geq m+1$. Next, we generalize a result of Erd\H{o}s and Mauldin, by constructing an infinite measure set $A\subseteq\mathbb{R}^n$ such that every $n$-parallelotope with vertices in $A$ has volume strictly smaller than $1$. Finally, some results complementing the literature on isometric embeddings of hypercube graphs and on the hyperbolic analogue of the Hadwiger--Nelson problem also follow as byproducts of our approaches.