### Abstract

We prove that coloring a 3-uniform 2-colorable hypergraph with c colors is NP-hard for any constant c. The best known algorithm [20] colors such a graph using O(n ^{1/5}) colors. Our result immediately implies that for any constants k ≥ 3 and c _{2} > c _{1} > 1, coloring a k-uniform c _{1}-colorable hypergraph with c _{2} colors is NP-hard; the case k = 2, however, remains wide open. This is the first hardness result for approximately-coloring a 3-uniform hypergraph that is colorable with a constant number of colors. For k ≥ 4 such a result has been shown by [14], who also discussed the inherent difference between the k = 3 case and k ≥ 4. Our proof presents a new connection between the Long-Code and the Kneser graph, and relies on the high chromatic numbers of the Kneser graph [19,22] and the Schrijver graph [26]. We prove a certain maximization variant of the Kneser conjecture, namely that any coloring of the Kneser graph by fewer colors than its chromatic number, has 'many' non-monochromatic edges.

Original language | English (US) |
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Pages (from-to) | 519-535 |

Number of pages | 17 |

Journal | Combinatorica |

Volume | 25 |

Issue number | 5 |

DOIs | |

State | Published - Sep 1 2005 |

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Computational Mathematics

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## Cite this

*Combinatorica*,

*25*(5), 519-535. https://doi.org/10.1007/s00493-005-0032-4