## Abstract

Let α _{1},...,α _{k} satisfy ∑ _{i}α _{i}=1 and suppose a k-uniform hypergraph on n vertices satisfies the following property; in any partition of its vertices into k sets A _{1},...,A _{k} of sizes α _{1}n,...,α _{k}n, the number of edges intersecting A _{1},...,A _{k} is (asymptotically) the number one would expect to find in a random k-uniform hypergraph. Can we then infer that H is quasi-random? We show that the answer is negative if and only if α _{1}= ... =α _{k}=1/k. This resolves an open problem raised in 1991 by Chung and Graham [J AMS 4 (1991), 151-196]. While hypergraphs satisfying the property corresponding to α _{1}= ... =α _{k}=1/k are not necessarily quasi-random, we manage to find a characterization of the hypergraphs satisfying this property. Somewhat surprisingly, it turns out that (essentially) there is a unique non quasi-random hypergraph satisfying this property. The proofs combine probabilistic and algebraic arguments with results from the theory of association schemes.

Original language | English |
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Pages (from-to) | 105-131 |

Number of pages | 27 |

Journal | Random Structures and Algorithms |

Volume | 40 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2012 |

Externally published | Yes |

## Keywords

- Cut properties
- Hypergraph
- Quasi-randomness