The Fractional Chromatic Number of the Plane

TL;DR

This paper investigates the fractional chromatic number of the plane, providing improved bounds that narrow the gap between known lower and upper limits, advancing understanding of this longstanding mathematical problem.

Contribution

It improves the lower bound of the fractional chromatic number of the plane from 3.5556 to approximately 3.6190, refining previous estimates.

Findings

Lower bound increased to 76/21 (~3.6190)

Previous bounds were 3.5556 to 4.3599

Advances understanding of 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 $76/21\approx3.6190$.