### 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 - May 1980 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics