The chromatic number of the plane with an interval of forbidden distances is at least 7

TL;DR

This paper proves that at least 7 colors are needed to color the plane so that points at distances within a small interval around 1 are differently colored, confirming a conjecture for Euclidean and Minkowski planes.

Contribution

It confirms G. Exoo's conjecture that at least 7 colors are necessary for the interval-based coloring problem in both Euclidean and Minkowski planes.

Findings

At least 7 colors are required for the interval coloring problem.

The conjecture holds for both Euclidean and Minkowski planes.

The result extends the understanding of the Hadwiger--Nelson problem variations.

Abstract

The work is devoted to one of the variations of the Hadwiger--Nelson problem on the chromatic number of the plane. In this formulation one needs to find for arbitrarily small $\varepsilon$ the least possible number of colors needed to color a Euclidean plane in such a way that any two points, the distance between which belongs to the interval $[1-\varepsilon, 1+\varepsilon]$, are colored differently. The conjecture proposed by G. Exoo in 2004, states that for arbitrary positive $\varepsilon$ at least 7 colors are required. Also, with a sufficiently small $\varepsilon$ the number of colors is exactly 7. The main result of the present paper is that the conjecture is true for the Euclidean plane as well as for any Minkowski plane.