### Abstract

Let ℋ be a 3-uniform hypergraph on an $n$-element vertex set $V$. The neighbourhood of $a,b\in V$ is $N(ab):= {x: abx ∈ (ℋ)} $. Such a 3-graph has independent neighbourhoods if no $N(ab)$ contains an edge of ℋ. This is equivalent to ℋ not containing a copy of double struck F sign3,2 :={abx, aby, abz, xyz}. In this paper we prove an analogue of the Andrásfai-Erdös-Sós theorem for triangle-free graphs with minimum degree exceeding $2n/5$. It is shown that any double struck F sign-free 3-graph with minimum degree exceeding $(4/9-1/125){n} {2} is bipartite, (for n > n_{0}), i.e., the vertices of ℋ can be split into two parts so that every triple meets both parts. This is, in fact, a Turán-type result. It solves a problem of Erdös and T.Sós, and answers a question of Mubayi and Rödl that ex(n,double struck F sign_{3,2})=max(α}(n-α)\left({α 2}. Here the right-hand side is 4/9{n}{3}+O(n^{2})$. Moreover e(ℋ)=ex(n,double struck F sign3,2) is possible only if V(ℋ)$ can be partitioned into two sets $A$ and $B$ so that each triple of ℋ intersects $A$ in exactly two vertices and $B$ in one.

Original language | English |
---|---|

Pages (from-to) | 795-813 |

Number of pages | 19 |

Journal | Combinatorics Probability and Computing |

Volume | 14 |

Issue number | 5-6 |

DOIs | |

Publication status | Published - Nov 2005 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics
- Mathematics(all)
- Discrete Mathematics and Combinatorics
- Statistics and Probability

### Cite this

*Combinatorics Probability and Computing*,

*14*(5-6), 795-813. https://doi.org/10.1017/S0963548305006905

**On triple systems with independent neighbourhoods.** / Füredi, Z.; Pikhurko, Oleg; Simonovits, M.

Research output: Contribution to journal › Article

*Combinatorics Probability and Computing*, vol. 14, no. 5-6, pp. 795-813. https://doi.org/10.1017/S0963548305006905

}

TY - JOUR

T1 - On triple systems with independent neighbourhoods

AU - Füredi, Z.

AU - Pikhurko, Oleg

AU - Simonovits, M.

PY - 2005/11

Y1 - 2005/11

N2 - Let ℋ be a 3-uniform hypergraph on an $n$-element vertex set $V$. The neighbourhood of $a,b\in V$ is $N(ab):= {x: abx ∈ (ℋ)} $. Such a 3-graph has independent neighbourhoods if no $N(ab)$ contains an edge of ℋ. This is equivalent to ℋ not containing a copy of double struck F sign3,2 :={abx, aby, abz, xyz}. In this paper we prove an analogue of the Andrásfai-Erdös-Sós theorem for triangle-free graphs with minimum degree exceeding $2n/5$. It is shown that any double struck F sign-free 3-graph with minimum degree exceeding $(4/9-1/125){n} {2} is bipartite, (for n > n0), i.e., the vertices of ℋ can be split into two parts so that every triple meets both parts. This is, in fact, a Turán-type result. It solves a problem of Erdös and T.Sós, and answers a question of Mubayi and Rödl that ex(n,double struck F sign3,2)=max(α}(n-α)\left({α 2}. Here the right-hand side is 4/9{n}{3}+O(n2)$. Moreover e(ℋ)=ex(n,double struck F sign3,2) is possible only if V(ℋ)$ can be partitioned into two sets $A$ and $B$ so that each triple of ℋ intersects $A$ in exactly two vertices and $B$ in one.

AB - Let ℋ be a 3-uniform hypergraph on an $n$-element vertex set $V$. The neighbourhood of $a,b\in V$ is $N(ab):= {x: abx ∈ (ℋ)} $. Such a 3-graph has independent neighbourhoods if no $N(ab)$ contains an edge of ℋ. This is equivalent to ℋ not containing a copy of double struck F sign3,2 :={abx, aby, abz, xyz}. In this paper we prove an analogue of the Andrásfai-Erdös-Sós theorem for triangle-free graphs with minimum degree exceeding $2n/5$. It is shown that any double struck F sign-free 3-graph with minimum degree exceeding $(4/9-1/125){n} {2} is bipartite, (for n > n0), i.e., the vertices of ℋ can be split into two parts so that every triple meets both parts. This is, in fact, a Turán-type result. It solves a problem of Erdös and T.Sós, and answers a question of Mubayi and Rödl that ex(n,double struck F sign3,2)=max(α}(n-α)\left({α 2}. Here the right-hand side is 4/9{n}{3}+O(n2)$. Moreover e(ℋ)=ex(n,double struck F sign3,2) is possible only if V(ℋ)$ can be partitioned into two sets $A$ and $B$ so that each triple of ℋ intersects $A$ in exactly two vertices and $B$ in one.

UR - http://www.scopus.com/inward/record.url?scp=26644460016&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=26644460016&partnerID=8YFLogxK

U2 - 10.1017/S0963548305006905

DO - 10.1017/S0963548305006905

M3 - Article

VL - 14

SP - 795

EP - 813

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

SN - 0963-5483

IS - 5-6

ER -