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Coloring and density theorems for configurations of a given volume

Vjekoslav Kovač

TL;DR

This work advances Euclidean Ramsey theory by establishing both negative colorings and positive density results for fixed-volume point configurations such as simplices, rectangles, and parallelotopes in Euclidean space. The authors develop a coherent harmonic-analytic framework, including a regularity-type decomposition, Gaussian smoothing, and compact-rotation arguments, to translate density assumptions into monochromatic configurations in fixed dimensions and to produce measurable-coloring variants. Key achievements include a 25-coloring of $\mathbb{R}^2$ avoiding monochromatic unit-area rectangles, density theorems guaranteeing large-volume boxes and right simplices in positive-density sets, and parallelotope-avoidance constructions in higher dimensions, complemented by hypercube and hyperbolic embedding results. The results illuminate when fixed-volume patterns must appear under positivity or density hypotheses and highlight intriguing open questions for parallelograms and more general rectangular configurations.

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 Erdős 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 Erdős 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.

Coloring and density theorems for configurations of a given volume

TL;DR

This work advances Euclidean Ramsey theory by establishing both negative colorings and positive density results for fixed-volume point configurations such as simplices, rectangles, and parallelotopes in Euclidean space. The authors develop a coherent harmonic-analytic framework, including a regularity-type decomposition, Gaussian smoothing, and compact-rotation arguments, to translate density assumptions into monochromatic configurations in fixed dimensions and to produce measurable-coloring variants. Key achievements include a 25-coloring of avoiding monochromatic unit-area rectangles, density theorems guaranteeing large-volume boxes and right simplices in positive-density sets, and parallelotope-avoidance constructions in higher dimensions, complemented by hypercube and hyperbolic embedding results. The results illuminate when fixed-volume patterns must appear under positivity or density hypotheses and highlight intriguing open questions for parallelograms and more general rectangular configurations.

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 , or in a single large measurable subset . 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 Erdős and Graham, by coloring the Euclidean plane in colors without creating monochromatic rectangles of unit area. More generally, we construct a finite coloring of the Euclidean space such that no color-class contains the vertices of any (possibly rotated) -dimensional rectangular box of volume . 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 , and we establish it under an additional assumption . Also, motivated by a question of Graham on reasonable bounds in his result on monochromatic axes-aligned right-angled -dimensional simplices, we establish its measurable coloring and density variants with polylogarithmic bounds, again in dimensions . Next, we generalize a result of Erdős and Mauldin, by constructing an infinite measure set such that every -parallelotope with vertices in has volume strictly smaller than . 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.
Paper Structure (19 sections, 22 theorems, 249 equations, 8 figures, 1 table)

This paper contains 19 sections, 22 theorems, 249 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Triangles and simplices
  3. Rectangles and rectangular boxes
  4. Parallelograms and parallelotopes
  5. Distance graphs in the plane
  6. Embeddings of hypercube graphs
  7. Hyperbolic embeddings
  8. Paper outline and the main ideas
  9. Rectangles: proof of Theorem \ref{['thm:colorR2']}
  10. Rectangular boxes: proof of Theorem \ref{['thm:colorRn']}
  11. Parallelograms: proof of Theorem \ref{['thm:negparallel']}
  12. Hypercube graphs: proof of Theorem \ref{['thm:hypercubes']}
  13. Parallelotopes: proof of Theorem \ref{['thm:parallel']}
  14. Notation and preliminaries from harmonic analysis
  15. Right simplices: proofs of Theorem \ref{['thm:simpldensity']} and Corollary \ref{['cor:simpldensity']}
  16. ...and 4 more sections

Key Result

Theorem 1

For every integer $m\geqslant2$ there exists a constant $C_m\in(0,\infty)$ with the following properties for every $n\geqslant m+1$. In both parts of the theorem, the $m$-simplex can be chosen such that $m-1$ of its edges from the right-angled vertex are parallel to the coordinate vectors $\mathbbm{e}_1,\ldots,\mathbbm{e}_{m-1}$, while the remaining edge is parallel to the linear span of $\mathbb

Figures (8)

  • Figure 1: Illustration of Theorem \ref{['thm:simpldensity']}.
  • Figure 2: Embedding of a $1$-skeleton of an $n$-box.
  • Figure 3: Illustration of Theorem \ref{['thm:spacetimedensity']}.
  • Figure 4: Coordinatization of a parallelogram.
  • Figure 5: The circle misses the squares.
  • ...and 3 more figures

Theorems & Definitions (44)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9: Bardestani and Mallahi-Karai BMK17
  • Theorem 10
  • ...and 34 more