### Abstract

In 1948, De Bruijn and Erdös proved that a finite linear space on ν points has at least ν lines, with equality occurring if and only if the space is either a near-pencil (all points but one collinear) or a projective plane. In this paper, we study finite linear spaces which are not near-pencils. We obtain a lower bound for the number of lines (as a function of the number of points) for such linear spaces. A finite linear space which meets this bound can be obtained provided a suitable projective plane exists. We then investigate the converse: can a finite linear space meeting the bound be embedded in a projective plane.

Original language | English |
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Pages (from-to) | 49-62 |

Number of pages | 14 |

Journal | Discrete Mathematics |

Volume | 47 |

Issue number | C |

DOIs | |

Publication status | Published - 1983 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

Erdös, P., Mullin, R. C., Sós, V. T., & Stinson, D. R. (1983). Finite linear spaces and projective planes.

*Discrete Mathematics*,*47*(C), 49-62. https://doi.org/10.1016/0012-365X(83)90071-7