### Abstract

Rogers [A note on coverings, Matematika 4 (1957) 1-6] proved, for a given closed convex body C in n-dimensional Euclidean space R^{n}, the existence of a covering for R^{n} by translates of C with density cn ln n for an absolute constant c. A few years later, Erdo{combining double acute accent}s and Rogers [Covering space with convex bodies, Acta Arith. 7 (1962) 281-285] obtained the existence of such a covering having not only low-density cn ln n but also low multiplicity c^{′} n ln n for an absolute constant c^{′}. In this paper, we give a simple proof of Erdo{combining double acute accent}s and Rogers' theorem using the Lovász Local Lemma. Furthermore, we apply the result to the chromatic number of the unit-distance graph under ℓ_{p}-norm.

Original language | English |
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Pages (from-to) | 4495-4500 |

Number of pages | 6 |

Journal | Discrete Mathematics |

Volume | 308 |

Issue number | 19 |

DOIs | |

Publication status | Published - Oct 6 2008 |

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### Keywords

- Chromatic number of the unit-distance graph
- Convex body
- Covering
- Lovász Local Lemma
- Rogers

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

### Cite this

*Discrete Mathematics*,

*308*(19), 4495-4500. https://doi.org/10.1016/j.disc.2007.08.048