The fractional chromatic number of the plane is at least 4

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 $4$. A fundamental ingredient of the proof is the notion of geometric fractional chromatic number $\chi_{gf}(\mathbb{R}^2)$ introduced recently by Ambrus et al. First, we establish that $\chi_f(\mathbb{R}^2)=\chi_{gf}(\mathbb{R}^2)$ by exploiting the amenability of the group of Euclidean transformations in dimension 2. Second, we provide a specific unit distance graph $G$ on 27 vertices such that $\chi_{gf}(G)=4$. We also provide a natural connection of $\chi_f(\mathbb{R}^2)$ to the maximal size of independent sets in finite unit distance graphs.