### Abstract

Let ℋ be a family of r-subsets of a finite set X. Set D(ℋ)= {Mathematical expression}|{E:x∈E∈ℋ}|, (maximum degree). We say that ℋ is intersecting if for any H, H′ ∈ ℋ we have H ∩H′ ≠ 0. In this case, obviously, D(ℋ)≧|ℋ|/r. According to a well-known conjecture D(ℋ)≧|ℋ|/(r-1+1/r). We prove a slightly stronger result. Let ℋ be an r-uniform, intersecting hypergraph. Then either it is a projective plane of order r-1, consequently D(ℋ)=|ℋ|/(r-1+1/r), or D(ℋ)≧|ℋ|/(r-1). This is a corollary to a more general theorem on not necessarily intersecting hypergraphs.

Original language | English |
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Pages (from-to) | 155-162 |

Number of pages | 8 |

Journal | Combinatorica |

Volume | 1 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 1981 |

### Fingerprint

### Keywords

- AMS subject classification (1980): 05C65, 05C35, 05B25

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Mathematics(all)
- Computational Mathematics

### Cite this

**Maximum degree and fractional matchings in uniform hypergraphs.** / Füredi, Z.

Research output: Contribution to journal › Article

*Combinatorica*, vol. 1, no. 2, pp. 155-162. https://doi.org/10.1007/BF02579271

}

TY - JOUR

T1 - Maximum degree and fractional matchings in uniform hypergraphs

AU - Füredi, Z.

PY - 1981/6

Y1 - 1981/6

N2 - Let ℋ be a family of r-subsets of a finite set X. Set D(ℋ)= {Mathematical expression}|{E:x∈E∈ℋ}|, (maximum degree). We say that ℋ is intersecting if for any H, H′ ∈ ℋ we have H ∩H′ ≠ 0. In this case, obviously, D(ℋ)≧|ℋ|/r. According to a well-known conjecture D(ℋ)≧|ℋ|/(r-1+1/r). We prove a slightly stronger result. Let ℋ be an r-uniform, intersecting hypergraph. Then either it is a projective plane of order r-1, consequently D(ℋ)=|ℋ|/(r-1+1/r), or D(ℋ)≧|ℋ|/(r-1). This is a corollary to a more general theorem on not necessarily intersecting hypergraphs.

AB - Let ℋ be a family of r-subsets of a finite set X. Set D(ℋ)= {Mathematical expression}|{E:x∈E∈ℋ}|, (maximum degree). We say that ℋ is intersecting if for any H, H′ ∈ ℋ we have H ∩H′ ≠ 0. In this case, obviously, D(ℋ)≧|ℋ|/r. According to a well-known conjecture D(ℋ)≧|ℋ|/(r-1+1/r). We prove a slightly stronger result. Let ℋ be an r-uniform, intersecting hypergraph. Then either it is a projective plane of order r-1, consequently D(ℋ)=|ℋ|/(r-1+1/r), or D(ℋ)≧|ℋ|/(r-1). This is a corollary to a more general theorem on not necessarily intersecting hypergraphs.

KW - AMS subject classification (1980): 05C65, 05C35, 05B25

UR - http://www.scopus.com/inward/record.url?scp=51249183638&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=51249183638&partnerID=8YFLogxK

U2 - 10.1007/BF02579271

DO - 10.1007/BF02579271

M3 - Article

VL - 1

SP - 155

EP - 162

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 2

ER -