Ramsey problems for graphs in Euclidean spaces and Cartesian powers

TL;DR

The paper develops a framework for Ramsey-type questions in Euclidean spaces for general graphs H, introducing χ_H(ℝ^n) and χ^ind_H(ℝ^n) as the minimal color-count avoiding monochromatic unit-copies of H. It develops a versatile Cartesian-powers approach, centered on zero G-slice density and hypercube Turán-type results, to derive broad Ramsey conclusions for large Cartesian powers, including forests, even cycles, and layered graphs. It establishes exact or sharp asymptotic behavior in several cases (e.g., forests; C_{2ℓ} with ℓ=4 or ℓ≥6; long odd cycles), proves induced variants, and proves a canonical-type result ensuring induced unit-copies that are monochromatic or rainbow in sufficiently large dimensions. The work also provides bounds, specific small-graph results in the plane and higher dimensions, and a range of open problems, including the elusive value of χ_{C_4}(ℝ^2) and extensions to infinite graphs and other norms.

Abstract

Given a graph $H$, let $χ_H(\mathbb{R}^n)$ be the smallest positive integer $r$ such that there exists an $r$-coloring of $\mathbb{R}^n$ with no monochromatic unit-copy of $H$, that is a set of $|V(H)|$ vertices of the same color such that any two vertices corresponding to an edge of $H$ are at distance one. This Ramsey-type function extends the famous Hadwiger--Nelson problem on the chromatic number $χ(\mathbb{R}^n)=χ_{K_2}(\mathbb{R}^n)$ of the space from a complete graph $K_2$ on two vertices to an arbitrary graph $H$. It also extends the classical Euclidean Ramsey problem for congruent monochromatic subsets to the family of those defined by a specific subset of unit distances. Among others, we show that $χ_H(\mathbb{R}^n)=χ(\mathbb{R}^n)$ for any even cycle $H$ of length $8$ or at least $12$ as well as for any forest and that $χ_H(\mathbb{R}^n)=\lceilχ(\mathbb{R}^n)/2\rceil$ for any sufficiently long odd cycle. Our main tools and results, which are of independent interest, establish that Cartesian powers enjoy Ramsey-type properties for graphs with favorable Turán-type characteristics, such as zero hypercube Turán density. In addition, we prove induced variants of these results, find bounds on $χ_H(\mathbb{R}^n)$ for growing dimensions $n$, and prove a canonical-type result. We conclude with many open problems. One of these is to determine $χ_{C_4}(\mathbb{R}^2)$, for a cycle $C_4$ on four vertices.

Figures (2)

• Figure 1: The staircase coloring of $\mathbb R^2$. Two red horizontal stairs are highlighted.
• Figure 2: (1) a $4$-coloring of $\mathbb R^2$ via hexagonal tiling, (2) a $4$-coloring of $\mathbb R^2$ via square tiling.

Theorems & Definitions (44)

• Theorem 1.1
• Theorem 1.2
• Proposition 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Proposition 1.7
• Lemma 1.8: De Bruijn--Erdős theorem debruijn
• Lemma 1.9: Horvat--Pisanski HP10
• Theorem 1.10