English

On the chromatic numbers of 3-dimensional slices

Combinatorics 2024-02-01 v2 Metric Geometry

Abstract

We prove that for an arbitrary ε>0\varepsilon > 0 holds χ(R3×[0,ε]6)10, \chi (\mathbb{R}^3 \times [0,\varepsilon]^6) \geq 10, where χ(M)\chi(M) stands for the chromatic number of an (infinite) graph with the vertex set MM and the edge set consists of pairs of monochromatic points at the distance 1 apart.

Keywords

Cite

@article{arxiv.2208.02230,
  title  = {On the chromatic numbers of 3-dimensional slices},
  author = {D. D. Cherkashin and A. J. Kanel-Belov and G. A. Strukov and V. A. Voronov},
  journal= {arXiv preprint arXiv:2208.02230},
  year   = {2024}
}