## Abstract

The neighborhood of a pair of vertices u, v in a triple system is the set of vertices w such that uvw is an edge. A triple system H is semi-bipartite if its vertex set contains a vertex subset X such that every edge of H intersects X in exactly two points. It is easy to see that if H is semi-bipartite, then the neighborhood of every pair of vertices in H is an independent set. We show a partial converse of this statement by proving that almost all triple systems with vertex sets [n] and independent neighborhoods are semi-bipartite. Our result can be viewed as an extension of the Erdo{double acute}s-Kleitman-Rothschild theorem to triple systems.The proof uses the Frankl-Rödl hypergraph regularity lemma, and stability theorems. Similar results have recently been proved for hypergraphs with various other local constraints.

Original language | English (US) |
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Pages (from-to) | 1494-1518 |

Number of pages | 25 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 118 |

Issue number | 4 |

DOIs | |

State | Published - May 2011 |

## Keywords

- Independent neighborhoods
- Semi-bipartite
- Speed of hypergraph property

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics