### Abstract

Let r be a positive integer. A finite family H of pairwise intersecting r-sets is a maximal clique of order r, if for any set A ∉ H, |A| ≤ r there exists a member E ε{lunate} H such that A ∩ E = {circled division slash}. For instance, a finite projective plane of order r - 1 is a maximal clique. Let N(r) denote the minimum number of sets in a maximal clique of order r. We prove N(r) ≤ 3 4r^{2} whenever a projective plane of order r 2 exists. This disproves the known conjecture N(r) ≥ r^{2} - r + 1.

Original language | English |
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Pages (from-to) | 282-289 |

Number of pages | 8 |

Journal | Journal of Combinatorial Theory, Series A |

Volume | 28 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1980 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

**On maximal intersecting families of finite sets.** / Füredi, Z.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory, Series A*, vol. 28, no. 3, pp. 282-289. https://doi.org/10.1016/0097-3165(80)90071-0

}

TY - JOUR

T1 - On maximal intersecting families of finite sets

AU - Füredi, Z.

PY - 1980

Y1 - 1980

N2 - Let r be a positive integer. A finite family H of pairwise intersecting r-sets is a maximal clique of order r, if for any set A ∉ H, |A| ≤ r there exists a member E ε{lunate} H such that A ∩ E = {circled division slash}. For instance, a finite projective plane of order r - 1 is a maximal clique. Let N(r) denote the minimum number of sets in a maximal clique of order r. We prove N(r) ≤ 3 4r2 whenever a projective plane of order r 2 exists. This disproves the known conjecture N(r) ≥ r2 - r + 1.

AB - Let r be a positive integer. A finite family H of pairwise intersecting r-sets is a maximal clique of order r, if for any set A ∉ H, |A| ≤ r there exists a member E ε{lunate} H such that A ∩ E = {circled division slash}. For instance, a finite projective plane of order r - 1 is a maximal clique. Let N(r) denote the minimum number of sets in a maximal clique of order r. We prove N(r) ≤ 3 4r2 whenever a projective plane of order r 2 exists. This disproves the known conjecture N(r) ≥ r2 - r + 1.

UR - http://www.scopus.com/inward/record.url?scp=4243880958&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4243880958&partnerID=8YFLogxK

U2 - 10.1016/0097-3165(80)90071-0

DO - 10.1016/0097-3165(80)90071-0

M3 - Article

VL - 28

SP - 282

EP - 289

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

IS - 3

ER -