## Combinatorics of Finite GeometriesCombinatorics of Finite Geometries is an introductory text on the combinatorial theory of finite geometry. Assuming only a basic knowledge of set theory and analysis, it provides a thorough review of the topic and leads the student to results at the frontiers of research. This book begins with an elementary combinatorial approach to finite geometries based on finite sets of points and lines, and moves into the classical work on affine and projective planes. Later, it addresses polar spaces, partial geometries, and generalized quadrangles. The revised edition contains an entirely new chapter on blocking sets in linear spaces, which highlights some of the most important applications of blocking sets--from the initial game-theoretic setting to their very recent use in cryptography. Extensive exercises at the end of each chapter insure the usefulness of this book for senior undergraduate and beginning graduate students. |

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### Contents

Preface | xi |

Preface to the first edition | xiii |

1 Nearlinear spaces | 1 |

2 Linear spaces | 23 |

3 Projective spaces | 41 |

4 Affine spaces | 67 |

5 Polar spaces | 89 |

6 Generalized quadrangles | 112 |

7 Partial geometries | 138 |

8 Blocking sets | 158 |

Bibliography | 176 |

190 | |

191 | |

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### Common terms and phrases

3-space absolute points adjacent afﬁne affine plane affine space antiregular axis blocking set central collineation centre construction contains contradiction Corollary deﬁne deﬁnition Desarguesian dimension direct sum distinct points dual space equation example exchange property exercise Fano plane Figure ﬁnd ﬁnite finite projective planes ﬁrst ﬁxed point follows Geom geometry with parameters Hence homothety implies integer intersection isomorphic isotropic lines lemma Let f ligure line regular linear function linear space lines meet Math matrix maximal linear subspace near-linear space non-collinear points number of elements number of lines number of points ovoid pairs of points parallel class parameters oz partial geometry Pasch’s axiom plane H plane of order point regular points and lines polar space projective hyperplane projective space Proof Prove quadrangle with parameters quadric reader satisﬁes satisfying set of points space of ﬁgure Steiner system strongly regular graph subgeometry subplane subquadrangle subset Suppose Thas theorem triad vector space