### Abstract

A large variety of problems and results in Extremal Set Theory deal with estimates on the size of a family of sets with some restrictions on the intersections of its members. Notable examples of such results, among others, are the celebrated theorems of Fischer, Ray-Chaudhuri-Wilson and Frankl-Wilson on set systems with restricted pairwise intersections. These also can be considered as estimates on binary codes with given distances. In this paper we obtain the following extension of some of these results when the restrictions apply to k-wise intersections, for k > 2. Let L be a subset of non-negative integers of size s and let k > 2. A family F of subsets of an n-element set is called k-wise L-intersecting if the cardinality of the intersection of any k distinct members in F belongs to L. We prove that, for any fixed k and s and sufficiently large n, the size of every k-wise L-intersecting family is bounded by A figure is presented. This result is asymptotically best possible. In addition, we show that for an extremal k-wise L-intersecting family, L consists of s consecutive integers. Our proof combines tools from linear algebra with some combinatorial arguments.

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

Pages (from-to) | 143-159 |

Number of pages | 17 |

Journal | Journal of Combinatorial Theory, Series A |

Volume | 105 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2004 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Journal of Combinatorial Theory, Series A*,

*105*(1), 143-159. https://doi.org/10.1016/j.jcta.2003.10.008

**Extremal set systems with restricted k-wise intersections.** / Füredi, Z.; Sudakov, Benny.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory, Series A*, vol. 105, no. 1, pp. 143-159. https://doi.org/10.1016/j.jcta.2003.10.008

}

TY - JOUR

T1 - Extremal set systems with restricted k-wise intersections

AU - Füredi, Z.

AU - Sudakov, Benny

PY - 2004/1

Y1 - 2004/1

N2 - A large variety of problems and results in Extremal Set Theory deal with estimates on the size of a family of sets with some restrictions on the intersections of its members. Notable examples of such results, among others, are the celebrated theorems of Fischer, Ray-Chaudhuri-Wilson and Frankl-Wilson on set systems with restricted pairwise intersections. These also can be considered as estimates on binary codes with given distances. In this paper we obtain the following extension of some of these results when the restrictions apply to k-wise intersections, for k > 2. Let L be a subset of non-negative integers of size s and let k > 2. A family F of subsets of an n-element set is called k-wise L-intersecting if the cardinality of the intersection of any k distinct members in F belongs to L. We prove that, for any fixed k and s and sufficiently large n, the size of every k-wise L-intersecting family is bounded by A figure is presented. This result is asymptotically best possible. In addition, we show that for an extremal k-wise L-intersecting family, L consists of s consecutive integers. Our proof combines tools from linear algebra with some combinatorial arguments.

AB - A large variety of problems and results in Extremal Set Theory deal with estimates on the size of a family of sets with some restrictions on the intersections of its members. Notable examples of such results, among others, are the celebrated theorems of Fischer, Ray-Chaudhuri-Wilson and Frankl-Wilson on set systems with restricted pairwise intersections. These also can be considered as estimates on binary codes with given distances. In this paper we obtain the following extension of some of these results when the restrictions apply to k-wise intersections, for k > 2. Let L be a subset of non-negative integers of size s and let k > 2. A family F of subsets of an n-element set is called k-wise L-intersecting if the cardinality of the intersection of any k distinct members in F belongs to L. We prove that, for any fixed k and s and sufficiently large n, the size of every k-wise L-intersecting family is bounded by A figure is presented. This result is asymptotically best possible. In addition, we show that for an extremal k-wise L-intersecting family, L consists of s consecutive integers. Our proof combines tools from linear algebra with some combinatorial arguments.

UR - http://www.scopus.com/inward/record.url?scp=1042303110&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=1042303110&partnerID=8YFLogxK

U2 - 10.1016/j.jcta.2003.10.008

DO - 10.1016/j.jcta.2003.10.008

M3 - Article

AN - SCOPUS:1042303110

VL - 105

SP - 143

EP - 159

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

IS - 1

ER -