On the density of sets of the Euclidean plane avoiding distance 1

TL;DR

This paper investigates the maximum density of measurable subsets of the Euclidean plane that avoid pairs of points exactly one unit apart, establishing upper and lower bounds related to the fractional chromatic number.

Contribution

It provides new bounds on the maximum density of sets avoiding distance 1 in the plane and links this to the fractional chromatic number of the plane.

Findings

Upper bound of 0.25647 for the density of sets avoiding distance 1

Lower bound of 3.8991 for the fractional chromatic number

Improves understanding of geometric and coloring properties of the plane

Abstract

A subset $A \subset \mathbb R^2$ is said to avoid distance $1$ if: $\forall x,y \in A, \left\| x-y \right\|_2 \neq 1.$ In this paper we study the number $m_1(\mathbb R^2)$ which is the supremum of the upper densities of measurable sets avoiding distance 1 in the Euclidean plane. Intuitively, $m_1(\mathbb R^2)$ represents the highest proportion of the plane that can be filled by a set avoiding distance 1. This parameter is related to the fractional chromatic number $\chi_f(\mathbb R^2)$ of the plane. We establish that $m_1(\mathbb R^2) \leq 0.25647$ and $\chi_f(\mathbb R^2) \geq 3.8991$.