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 a 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.