### Abstract

A hypergraph ([n],E) is 3-color critical if it is not 2-colorable, but for all E∈E the hypergraph ([n],E\{E}) is 2-colorable. Lovász proved in 1976, that {pipe}E{pipe}≤nk-1 if E is k-uniform. Here we give a new algebraic proof and an ordered version that is a sharpening of Lovász' result.Let E⊆[n]k be a k-uniform set system on an underlying set [n] of n elements. Let us fix an ordering E _{1}, E _{2}, . . .E _{t} of E and a prescribed partition {A _{i}, B _{i}} of each E _{i} (i.e., A _{i}∪B _{i}=E _{i} and A _{i}∩B _{i}=∅). Assume that for all i=1, 2, . . ., t there exists a partition {C _{i}, D _{i}} of [n] such that E _{i}∩C _{i}=A _{i} and E _{i}∩D _{i}=B _{i}, but {E _{j}∩C _{i}, E _{j}∩D _{i}}≠{A _{j}, B _{j}} for all j<i. That is, the ith partition cuts the ith set as it was prescribed, but it does not cut any earlier set properly. Then t≤f(n,k):=n-1k-1+n-1k-2+⋯+n-10. This is sharp for k=2, 3. We show that this upper bound is almost the best possible, at least the first three terms are correct; we give constructions of size f(n, k)-O(n ^{k-4}) (for k fixed and n→∞). We also give constructions of sizes nk-1 for all n and k.Furthermore, in the 3-color-critical case (i.e. {A _{i}, B _{i}}={E _{i}, ∅} for all i), t≤nk-1.

Original language | English |
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Pages (from-to) | 844-852 |

Number of pages | 9 |

Journal | European Journal of Combinatorics |

Volume | 33 |

Issue number | 5 |

DOIs | |

Publication status | Published - Jul 1 2012 |

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

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## Cite this

*European Journal of Combinatorics*,

*33*(5), 844-852. https://doi.org/10.1016/j.ejc.2011.09.023