## Abstract

For a fixed k-uniform hypergraph D (k-graph for short, k ≥ 3), we say that a k-graph H satisfies property P_{D} (or property P^{*} _{D}) if it contains no copy (or no induced copy) of D. Our goal in this paper is to classify the k-graphs D for which there are property-testers for testing P_{D} and P^{*} _{D} whose query complexity is polynomial in 1/ε. For such k-graphs we say that property P_{D} (or property P^{*} _{D}) is easily testable. For P^{*} _{D}, we prove that aside from a single 3-graph, P^{*} _{D} is easily testable if and only if D is a single k-edge. We further show that for large k, one can use more sophisticated techniques in order to obtain better lower bounds for any large enough k-graph. These results extend and improve the authors’ previous results about graphs (SODA 2004) and results by Kohayakawa, Nagle and Rödl on k-graphs (ICALP 2002). For P_{D}, we show that for any k-partite k-graph D, property P_{D} is easily testable. This is established by giving an efficient one-sided-error property-tester for P_{D}, which improves the one obtained by Kohayakawa et al. We further prove a nearly matching lower bound on the query complexity of such a property-tester. Finally, we give a sufficient condition for inferring that P_{D} is not easily testable. Though our results do not supply a complete characterization of the k-graphs for which P_{D} is easily testable, they are a natural extension of the previous results about graphs (Alon, 2002). Our proofs combine results and arguments from additive number theory, linear algebra, and extremal hypergraph theory. We also develop new techniques, which we believe are of independent interest. The first is a construction of a dense set of integers which does not contain a subset that satisfies a certain set of linear equations. The second is an algebraic construction of certain extremal hypergraphs. These techniques have already been applied in two papers under publication by the authors.

Original language | English (US) |
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Pages (from-to) | 177-216 |

Number of pages | 40 |

Journal | Theory of Computing |

Volume | 1 |

DOIs | |

State | Published - Oct 12 2005 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computational Theory and Mathematics

## Keywords

- Additive number theory
- Extremal problems
- Hypergraphs
- Linear algebra
- Lower bounds
- Property testing