The fractional chromatic number of the plane is at least 4

TL;DR

The paper resolves a long-standing question about the unit-distance graph of the plane by proving $χ_f(\mathbb{R}^2)\ge 4$ through the geometric fractional chromatic number $χ_{gf}$ and a finite graph $G_{27}$ with $χ_{gf}(G_{27})=4$. It establishes that the finitary fractional chromatic number and the finitary Hall ratio of the plane coincide, $χ_{f,0}(\mathbb{R}^2)=χ_{gf,0}(\mathbb{R}^2)=ρ(\mathbb{R}^2)$, and hence $χ_f(\mathbb{R}^2)=ρ(\mathbb{R}^2)=1/α_1(\mathbb{R}^2)$. A key methodological advance is a blow-up construction in 2D leveraging amenability to transfer high geometric fractional values to larger graphs, while a discrete-cube averaging argument yields the finitary bound. The results connect fractional chromatic numbers to Hall ratios and 1-avoiding set densities, with implications for density bounds and potential separations between measurable and finitary parameters in the plane.

Abstract

We prove that the fractional chromatic number $χ_f(\mathbb R^2)$ of the unit distance graph of the Euclidean plane is greater than or equal to $4$. Interestingly, however, we cannot present a finite subgraph $G$ of the plane such that $χ_f(G)\ge 4$. Instead, we utilize the concept of the geometric fractional chromatic number $χ_{gf}(G)$, which was introduced recently in connection with density bounds for 1-avoiding sets. First, as $G$ ranges over finite subgraphs of the plane, we establish that the supremum of $χ_f(G)$ is the same as that of $χ_{gf}(G)$. The proof exploits the amenability of the group of Euclidean transformations in dimension 2 and, as such, we do not know whether the analogous statement holds in higher dimensions. We then present a specific planar unit distance graph $G$ on 27 vertices such that $χ_{gf}(G)=4$, and conclude $χ_f(\mathbb R^2)\ge 4$ as a corollary. As another main result we show that the finitary fractional chromatic number and the Hall ratio of the plane are equal. As a consequence, we conclude that there exist finite unit distance graphs with independence ratio $\frac{1}{4}+\varepsilon$, while we conjecture that the value $\frac{1}{4}$ cannot be reached.

Figures (2)

• Figure 1: The unit distance graph $G_{27}$.
• Figure 2: Plot of the highest $\chi_{gf}(G)$ value encountered among $n$-vertex graphs $G$, for each value of $n$.

Theorems & Definitions (14)

• Theorem 1
• proof
• Theorem 2
• proof
• Remark 1
• Theorem 3
• proof
• Corollary 1
• proof
• Conjecture 1