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

Vsevolod Voronov

TL;DR

This work resolves a key case of the Hadwiger-Nelson-type problem by proving that the plane requires at least 7 colors to avoid all distances in the interval $[1-\varepsilon,1+\varepsilon]$ for arbitrarily small $\varepsilon$. The authors develop a framework based on strictly convex norms, discrete hexagonal colorings, and a cycle of geometric constructions to show that 6-colorings fail: any such coloring yields a configuration (the bicycle) that cannot be properly colored with 3 colors. The central argument tracks trichromatic points and unit-circle colorings, culminating in a contradiction that persists under norm relaxation from Euclidean to Minkowski planes. This establishes the conjecture for both the Euclidean plane and any 2D normed (Minkowski) plane and suggests broader applicability to related distance-avoidance problems in planar geometry.

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.

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

TL;DR

This work resolves a key case of the Hadwiger-Nelson-type problem by proving that the plane requires at least 7 colors to avoid all distances in the interval for arbitrarily small . The authors develop a framework based on strictly convex norms, discrete hexagonal colorings, and a cycle of geometric constructions to show that 6-colorings fail: any such coloring yields a configuration (the bicycle) that cannot be properly colored with 3 colors. The central argument tracks trichromatic points and unit-circle colorings, culminating in a contradiction that persists under norm relaxation from Euclidean to Minkowski planes. This establishes the conjecture for both the Euclidean plane and any 2D normed (Minkowski) plane and suggests broader applicability to related distance-avoidance problems in planar geometry.

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 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 , are colored differently. The conjecture proposed by G. Exoo in 2004, states that for arbitrary positive at least 7 colors are required. Also, with a sufficiently small 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.
Paper Structure (7 sections, 17 theorems, 43 equations, 6 figures)

This paper contains 7 sections, 17 theorems, 43 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Colorings of the circle
  4. Coloring of complementary arcs
  5. Chromatic number of the bicycle
  6. Proof of the main result
  7. Conclusion

Key Result

Theorem 1.1

For any 2-dimensional norm $\|\cdot\|_U$ and $\varepsilon>0$ it holds that

Figures (6)

  • Figure 1: 7-coloring of the plane
  • Figure 2: Construction of 9-gon in Proposition \ref{['prop_9gon']}
  • Figure 3: A pair of complementary arcs colored in $1, 2, 3$
  • Figure 4: Admissible transitions between pairs of colors
  • Figure 5: Complementary arcs on the bicycle
  • ...and 1 more figures

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.1
  • Theorem 1.3
  • Corollary 2.1: martini2001geometry, p. 114.
  • Corollary 2.2
  • proof
  • Lemma 2.1: follows from ahmadi2019polynomial, Theorem 3.1
  • Definition 2.1
  • Proposition 2.1
  • ...and 33 more