Euclidean Red–Blue Ramsey Sets

Definitions and mathematical statement

For a positive integer $n$, identify Euclidean space $\mathbb E^n$ with ${\mathbb{R}}^n$, equipped with the distance $d(x,y)=\|x-y\|_2$. A red--blue coloring is an arbitrary function $c:{\mathbb{R}}^n\to\{\mathrm{red},\mathrm{blue}\}$; no measurability or other regularity is required. For a subset $K\subseteq{\mathbb{R}}^n$, a blue isometric copy of $K$ means the image of a map $f:K\to{\mathbb{R}}^n$ satisfying \[ \|f(u)-f(v)\|_2=\|u-v\|_2\quad\text{for all }u,v\in K, \qquad c(f(u))=\mathrm{blue}\quad\text{for all }u\in K. \] Such a map is injective. Determine all pairs $(n,K)$, with $n\geq 1$ and $K\subseteq{\mathbb{R}}^n$, for which \[ \begin{aligned} \forall c:{\mathbb{R}}^n\to\{\mathrm{red},\mathrm{blue}\},\qquad &\bigl[\exists x,y\in{\mathbb{R}}^n:\ \|x-y\|_2=1 \ \land\ c(x)=c(y)=\mathrm{red}\bigr]\\ &\quad\lor\ \bigl[\exists f:K\to{\mathbb{R}}^n:\ (\forall u,v\in K,\ \|f(u)-f(v)\|_2=\|u-v\|_2)\\ &\hspace{110pt}\land\ (\forall u\in K,\ c(f(u))=\mathrm{blue})\bigr]. \end{aligned} \] Equivalently, for each fixed ambient dimension, classify the subsets with this property. The supplied statement imposes no finiteness, boundedness, closedness, or nonemptiness condition on $K$, so arbitrary subsets, including the empty set and singletons, are retained. For the empty set, the displayed quantifiers use the unique empty map. The source does not explicitly specify the range of $n$; positive integer dimensions are used here, and whether dimension zero was intended remains unspecified.

Short English statement

Classify the subsets $K\subseteq{\mathbb{R}}^n$ for which every red--blue coloring of the ambient space contains either two red points exactly one unit apart or a blue copy of $K$ preserving all distances. The source labels the general classification open; this is supplied dated curation, not an independently verified status assessment.

Sources

• Supplied local registry record: permanent UnsolvedMath ID $30004353$, title "Euclidean Red--Blue Ramsey Sets," Oberwolfach Reports Open Problems collection, exact source code catalogue entry. No individual proposer is supplied; the proposed year is recorded as 2020.
• Supplied report citation: Combinatorics (2021), Oberwolfach Reports, DOI 10.4171/owr/2020/1, pp. 6--89. https://doi.org/10.4171/owr/2020/1. The background describes these reports as documenting mathematical workshops and their open-problem discussions. The supplied citation is retained without reconciling its year with the DOI and proposed-year metadata.
• Supplied literature reference: Conlon--Fox, "lines in Euclidean Ramsey theory." https://doi.org/10.1007/s00454-018-9980-5.
• Supplied literature reference: "recent bounds for Euclidean Ramsey theory." https://arxiv.org/abs/2606.17194.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the general classification of pairs $(n,K)$ open, reports strong exponential lower-bound constructions by Conlon--Fox, and reports that many small-dimensional cases are known without resolving the full question. This is supplied dated curation derived from the OWR record, not independently verified literature assessment. Its time-sensitive status and cited claims require independent verification.