### Abstract

For a finite projective plane ∏, let χ̄(∏) denote the maximum number of classes in a partition of the point set, such that each line has at least two points in the same partition class. We prove that the best possible general estimate in terms of the order of projective planes is q ^{2} - q - ⊖( √ q), which is tight apart from a multiplicative constant in the third term √ q: (1) As q→∞, χ̄( ∏) ≤ q^{2} - q - √ q/^{2} + o( √ q) holds for every projective plane ∏ of order q. (2) If q is a square, then the Galois plane of order q satisfies χ̄(PG(2, q)) ≥ q^{2} - q ^{- 2} √ q. Our results asymptotically solve a ten-year-old open problem in the coloring theory of mixed hypergraphs, where χ̄( ∏) is termed the upper chromatic number of ∏. Further improvements on the upper bound (1) are presented for Galois planes and their subclasses.

Original language | English |
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Pages (from-to) | 221-230 |

Number of pages | 10 |

Journal | Journal of Combinatorial Designs |

Volume | 16 |

Issue number | 3 |

DOIs | |

Publication status | Published - May 1 2008 |

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### Keywords

- 05B25
- 05C15
- Finite projective plane
- Hypergraph
- MSC 2000
- Upper chromatic number
- Vertex coloring

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

### Cite this

*Journal of Combinatorial Designs*,

*16*(3), 221-230. https://doi.org/10.1002/jcd.20169