On maximal intersecting families of finite sets

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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.

Original languageEnglish
Pages (from-to)282-289
Number of pages8
JournalJournal of Combinatorial Theory, Series A
Issue number3
Publication statusPublished - May 1980


ASJC Scopus subject areas

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

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