The Fractional Chromatic Number of the Plane

Abstract

The chromatic number of the plane is the chromatic number of the uncountably infinite graph that has as its vertices the points of the plane and has an edge between two points if their distance is 1. This chromatic number is denoted $\chi(\mathcal{R}^2)$. The problem was introduced in 1950, and shortly thereafter it was proved that $4\le \chi(\mathcal{R}^2)\le 7$. These bounds are both easy to prove, but after more than 60 years they are still the best known. In this paper, we investigate $\chi_f(\mathcal{R}^2)$, the fractional chromatic number of the plane. The previous best bounds (rounded to five decimal places) were $3.5556 \le \chi_f(\mathcal{R}^2)\le 4.3599$. Here we improve the lower bound to $76/21\approx3.6190$.