### Abstract

We asymptotically solve an open problem raised independently by Sterboul (Colloq Math Soc J Bolyai 3:1387-1404, 1973), Arocha et al. (J Graph Theory 16:319-326, 1992) and Voloshin (Australas J Combin 11:25-45, 1995). For integers n ≥ k ≥ 2, let f(n, k) denote the minimum cardinality of a famil H of k-element sets over an n-element underlying set X such that every partition X_{1} ∪. . .∪ X_{k} = X into k nonempty classes completely partitions some H ∈ H; that is, {pipe}H ∩ X_{i}{pipe} = 1 holds for all 1 ≤ i ≤ k. This very natural function-whose defining property for k = 2 just means that H is a connected graph-turns out to be related to several extensively studied areas in combinatorics and graph theory. We prove general estimates from which, follows for every fixed k, and also for all k = o(n^{1/3}), as n → ∞. Further, we disprove a conjecture of Arocha et al. (1992). The exact determination of f(n,k) for all n and k appears to be far beyond reach to our present knowledge, since e.g. the equality, holds, where the last term is the Turán number for graphs of girth 5.

Original language | English |
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Pages (from-to) | 807-816 |

Number of pages | 10 |

Journal | Graphs and Combinatorics |

Volume | 25 |

Issue number | 6 |

DOIs | |

Publication status | Published - Mar 1 2010 |

### Keywords

- C-coloring
- Cochromatic number
- Crossing set
- Hypergraph
- Partition
- Upper chromatic number
- Vertex coloring

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

*Graphs and Combinatorics*,

*25*(6), 807-816. https://doi.org/10.1007/s00373-010-0890-4