  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.
