### 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 |
---|---|

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

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

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

**Covering the n-space by convex bodies and its chromatic number.** / Füredi, Z.; Kang, J. H.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 308, no. 19, pp. 4495-4500. https://doi.org/10.1016/j.disc.2007.08.048

}

TY - JOUR

T1 - Covering the n-space by convex bodies and its chromatic number

AU - Füredi, Z.

AU - Kang, J. H.

PY - 2008/10/6

Y1 - 2008/10/6

N2 - Rogers [A note on coverings, Matematika 4 (1957) 1-6] proved, for a given closed convex body C in n-dimensional Euclidean space Rn, the existence of a covering for Rn 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 ℓp-norm.

AB - Rogers [A note on coverings, Matematika 4 (1957) 1-6] proved, for a given closed convex body C in n-dimensional Euclidean space Rn, the existence of a covering for Rn 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 ℓp-norm.

KW - Chromatic number of the unit-distance graph

KW - Convex body

KW - Covering

KW - Lovász Local Lemma

KW - Rogers

UR - http://www.scopus.com/inward/record.url?scp=45449105030&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=45449105030&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2007.08.048

DO - 10.1016/j.disc.2007.08.048

M3 - Article

AN - SCOPUS:45449105030

VL - 308

SP - 4495

EP - 4500

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 19

ER -