### Abstract

A Berge-K
_{4}
in a triple system is a configuration with four vertices v1, v2, v3, v4 and six distinct triples { eij: 1 ≤ i < j ≤ 4} such that { vi, vj} ⊂ eij for every 1 ≤ i < j ≤ 4. We denote by B the set of Berge-K
_{4}
configurations. A triple system is B-free if it does not contain any member of B. We prove that the maximum number of triples in a B -free triple system on n > 6 points is obtained by the balanced complete 3-partite triple system: all triples { abc : a ∈ A, b ∈ B, c ∈ C} where A,B,C is a partition of n points with ⌊ n/3⌋ = | A| ≤ |B| ≤ | C| = ⌈ n/3⌉.

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

Pages (from-to) | 383-392 |

Number of pages | 10 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 33 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 2019 |

### Fingerprint

### Keywords

- Berge-G hypergraphs
- Triple system
- Turán number

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**
The turán number of berge-K
_{4}
in triple systems
.** / Gyárfás, A.

Research output: Contribution to journal › Article

_{4}in triple systems ',

*SIAM Journal on Discrete Mathematics*, vol. 33, no. 1, pp. 383-392. https://doi.org/10.1137/18M1204048

}

TY - JOUR

T1 - The turán number of berge-K 4 in triple systems

AU - Gyárfás, A.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - A Berge-K 4 in a triple system is a configuration with four vertices v1, v2, v3, v4 and six distinct triples { eij: 1 ≤ i < j ≤ 4} such that { vi, vj} ⊂ eij for every 1 ≤ i < j ≤ 4. We denote by B the set of Berge-K 4 configurations. A triple system is B-free if it does not contain any member of B. We prove that the maximum number of triples in a B -free triple system on n > 6 points is obtained by the balanced complete 3-partite triple system: all triples { abc : a ∈ A, b ∈ B, c ∈ C} where A,B,C is a partition of n points with ⌊ n/3⌋ = | A| ≤ |B| ≤ | C| = ⌈ n/3⌉.

AB - A Berge-K 4 in a triple system is a configuration with four vertices v1, v2, v3, v4 and six distinct triples { eij: 1 ≤ i < j ≤ 4} such that { vi, vj} ⊂ eij for every 1 ≤ i < j ≤ 4. We denote by B the set of Berge-K 4 configurations. A triple system is B-free if it does not contain any member of B. We prove that the maximum number of triples in a B -free triple system on n > 6 points is obtained by the balanced complete 3-partite triple system: all triples { abc : a ∈ A, b ∈ B, c ∈ C} where A,B,C is a partition of n points with ⌊ n/3⌋ = | A| ≤ |B| ≤ | C| = ⌈ n/3⌉.

KW - Berge-G hypergraphs

KW - Triple system

KW - Turán number

UR - http://www.scopus.com/inward/record.url?scp=85064397142&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85064397142&partnerID=8YFLogxK

U2 - 10.1137/18M1204048

DO - 10.1137/18M1204048

M3 - Article

VL - 33

SP - 383

EP - 392

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 1

ER -