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On the chromatic numbers of 3-dimensional slices

D. D. Cherkashin, A. J. Kanel-Belov, G. A. Strukov, V. A. Voronov

Abstract

We prove that for an arbitrary $\varepsilon > 0$ holds \[ χ(\mathbb{R}^3 \times [0,\varepsilon]^6) \geq 10, \] where $χ(M)$ stands for the chromatic number of an (infinite) graph with the vertex set $M$ and the edge set consists of pairs of monochromatic points at the distance 1 apart.

On the chromatic numbers of 3-dimensional slices

Abstract

We prove that for an arbitrary holds \[ χ(\mathbb{R}^3 \times [0,\varepsilon]^6) \geq 10, \] where stands for the chromatic number of an (infinite) graph with the vertex set and the edge set consists of pairs of monochromatic points at the distance 1 apart.
Paper Structure (22 sections, 9 theorems, 66 equations, 4 figures)

This paper contains 22 sections, 9 theorems, 66 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Nelson--Hadwiger problem and its planar generalizations
  3. Structure of the paper.
  4. The chromatic numbers of real 3-dimensional slices
  5. The chromatic numbers of 2-dimensional rational slices
  6. Notation and auxiliary lemmas
  7. Proof of Theorem \ref{['point_c4']}
  8. (i)
  9. (ii)
  10. (iii)
  11. (iv)
  12. Proof of the main result
  13. Outline of the proof.
  14. Step 1.
  15. Step 2.
  16. ...and 7 more sections

Key Result

Theorem 1

For every positive $\varepsilon$ holds

Figures (4)

  • Figure 1: A path of length four between $u$ and $v_4$.
  • Figure 2: Illustration to item (ii).
  • Figure 3: Illustration to item (iv). The construction of sets $D'_{ik}$
  • Figure 4: Construction of a rainbow 10-point set.

Theorems & Definitions (15)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Proposition 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1: Knaster--Kuratowski--Mazurkiewicz
  • Lemma 2
  • ...and 5 more