### Abstract

We investigate the upper chromatic number of the hypergraph formed by the points and the k-dimensional subspaces of PG(n, q); that is, the most number of colors that can be used to color the points so that every k-subspace contains at least two points of the same color. Clearly, if one colors the points of a double blocking set with the same color, the rest of the points may get mutually distinct colors. This gives a trivial lower bound, and we prove that it is sharp in many cases. Due to this relation with double blocking sets, we also prove that for t ≤ 3/8p + 1, a small t-fold (weighted) (n − k)-blocking set of PG(n, p), p prime, must contain the weighted sum of t not necessarily distinct (n − k)-spaces.

Original language | English |
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Pages (from-to) | 118-140 |

Number of pages | 23 |

Journal | Journal of Combinatorial Designs |

Volume | 28 |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 1 2020 |

### Keywords

- multiple weigthed blocking sets
- projective spaces
- upper chromatic number

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

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## Cite this

*Journal of Combinatorial Designs*,

*28*(2), 118-140. https://doi.org/10.1002/jcd.21686