### Abstract

We study the function b(n, d), the maximal number of atoms defined by n d-dimensional boxes, i.e. parallelopipeds in the d-dimensional Euclidean space with sides parallel to the coordinate axes. We characterize extremal interval families defining b(n, 1)=2 n-1 atoms and we show that b(n, 2)=2 n^{ 2}-6 n+7. We prove that for every d, {Mathematical expression} exists and {Mathematical expression}. Moreover, we obtain b*(3)=8/9.

Original language | English |
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Pages (from-to) | 193-204 |

Number of pages | 12 |

Journal | Combinatorica |

Volume | 5 |

Issue number | 3 |

DOIs | |

Publication status | Published - Sep 1 1985 |

### Keywords

- AMS subject classification (1980): 51M05, 52A20

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Computational Mathematics

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## Cite this

Gyárfás, A., Lehel, J., & Tuza, ZS. (1985). How many atoms can be defined by boxes?

*Combinatorica*,*5*(3), 193-204. https://doi.org/10.1007/BF02579362