### 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 |
---|---|

Pages (from-to) | 49-62 |

Number of pages | 14 |

Journal | Discrete Mathematics |

Volume | 47 |

Issue number | C |

DOIs | |

Publication status | Published - 1983 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

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

**Finite linear spaces and projective planes.** / Erdős, P.; Mullin, R. C.; Sós, V. T.; Stinson, D. R.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 47, no. C, pp. 49-62. https://doi.org/10.1016/0012-365X(83)90071-7

}

TY - JOUR

T1 - Finite linear spaces and projective planes

AU - Erdős, P.

AU - Mullin, R. C.

AU - Sós, V. T.

AU - Stinson, D. R.

PY - 1983

Y1 - 1983

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0039582863&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0039582863&partnerID=8YFLogxK

U2 - 10.1016/0012-365X(83)90071-7

DO - 10.1016/0012-365X(83)90071-7

M3 - Article

VL - 47

SP - 49

EP - 62

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - C

ER -