### Abstract

One of the central problems of extremal hypergraph theory is the description of unavoidable subhypergraphs, in other words, the Turan problem. Let a = (a_{1}, ..., a_{p}) be a sequence of positive integers, p ≥ 2, k = a_{1} + ... + a_{p}. An a-cluster is a family of k-sets {F_{0}, ..., F_{p}} such that the sets F_{i} \ F_{0} are pairwise disjoint (1 ≤ i ≤ p), | F_{i} \ F_{0} | = a_{i}, and the sets F_{0} \ F_{i} are pairwise disjoint, too. Given a there is a unique a-cluster, and the sets F_{0} \ F_{i} form an a-partition of F_{0}. With an intensive use of the delta-system method we prove that for k > p > 1 and sufficiently large n, (n > n_{0} (k)), if F is an n-vertex k-uniform family with | F | exceeding the Erdo{combining double acute accent}s-Ko-Rado bound ((n - 1; k - 1)), then F contains an a-cluster. The only extremal family consists of all the k-subsets containing a given element.

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

Pages (from-to) | 63-67 |

Number of pages | 5 |

Journal | Electronic Notes in Discrete Mathematics |

Volume | 34 |

DOIs | |

Publication status | Published - Aug 1 2009 |

### Fingerprint

### Keywords

- Erdo{combining double acute accent}s-Ko-Rado
- set systems
- traces

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Electronic Notes in Discrete Mathematics*,

*34*, 63-67. https://doi.org/10.1016/j.endm.2009.07.011

**Unavoidable subhypergraphs : a-clusters.** / Füredi, Z.; Özkahya, Lale.

Research output: Contribution to journal › Article

*Electronic Notes in Discrete Mathematics*, vol. 34, pp. 63-67. https://doi.org/10.1016/j.endm.2009.07.011

}

TY - JOUR

T1 - Unavoidable subhypergraphs

T2 - a-clusters

AU - Füredi, Z.

AU - Özkahya, Lale

PY - 2009/8/1

Y1 - 2009/8/1

N2 - One of the central problems of extremal hypergraph theory is the description of unavoidable subhypergraphs, in other words, the Turan problem. Let a = (a1, ..., ap) be a sequence of positive integers, p ≥ 2, k = a1 + ... + ap. An a-cluster is a family of k-sets {F0, ..., Fp} such that the sets Fi \ F0 are pairwise disjoint (1 ≤ i ≤ p), | Fi \ F0 | = ai, and the sets F0 \ Fi are pairwise disjoint, too. Given a there is a unique a-cluster, and the sets F0 \ Fi form an a-partition of F0. With an intensive use of the delta-system method we prove that for k > p > 1 and sufficiently large n, (n > n0 (k)), if F is an n-vertex k-uniform family with | F | exceeding the Erdo{combining double acute accent}s-Ko-Rado bound ((n - 1; k - 1)), then F contains an a-cluster. The only extremal family consists of all the k-subsets containing a given element.

AB - One of the central problems of extremal hypergraph theory is the description of unavoidable subhypergraphs, in other words, the Turan problem. Let a = (a1, ..., ap) be a sequence of positive integers, p ≥ 2, k = a1 + ... + ap. An a-cluster is a family of k-sets {F0, ..., Fp} such that the sets Fi \ F0 are pairwise disjoint (1 ≤ i ≤ p), | Fi \ F0 | = ai, and the sets F0 \ Fi are pairwise disjoint, too. Given a there is a unique a-cluster, and the sets F0 \ Fi form an a-partition of F0. With an intensive use of the delta-system method we prove that for k > p > 1 and sufficiently large n, (n > n0 (k)), if F is an n-vertex k-uniform family with | F | exceeding the Erdo{combining double acute accent}s-Ko-Rado bound ((n - 1; k - 1)), then F contains an a-cluster. The only extremal family consists of all the k-subsets containing a given element.

KW - Erdo{combining double acute accent}s-Ko-Rado

KW - set systems

KW - traces

UR - http://www.scopus.com/inward/record.url?scp=67749147457&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=67749147457&partnerID=8YFLogxK

U2 - 10.1016/j.endm.2009.07.011

DO - 10.1016/j.endm.2009.07.011

M3 - Article

VL - 34

SP - 63

EP - 67

JO - Electronic Notes in Discrete Mathematics

JF - Electronic Notes in Discrete Mathematics

SN - 1571-0653

ER -