The fractional chromatic number of the plane is at least 4

Abstract

We prove that the fractional chromatic number ${\chi }_f({\mathbb{R}}^2)$ 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 ${\chi }_f(G)\ge 4$. Instead, we utilize the concept of the geometric fractional chromatic number ${\chi }_{gf}(G)$, 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 ${\chi }_f(G)$ is the same as that of ${\chi }_{gf}(G)$. 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 ${\chi }_{gf}(G)=4$, and conclude ${\chi }_f({\mathbb{R}}^2)\ge 4$ 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 $\frac{1}{4}+\epsilon$, while we conjecture that the value $\frac{1}{4}$ cannot be reached.