Linear Spaces with Few Lines

1. Front Matter

   Pages I-XII

   PDF

2. Definition and basic properties of linear spaces

   • Klaus Metsch

   Pages 1-8

3. Lower bounds for the number of lines

   • Klaus Metsch

   Pages 9-14

4. Basic properties and results of (n+1,1)-designs

   • Klaus Metsch

   Pages 15-20

5. Points of degree n

   • Klaus Metsch

   Pages 21-30

6. Linear spaces with few lines

   • Klaus Metsch

   Pages 31-42

7. Embedding (n+1,1)-designs into projective planes

   • Klaus Metsch

   Pages 43-60

8. An optimal bound for embedding linear spaces into projective planes

   • Klaus Metsch

   Pages 61-73

9. The theorem of totten

   • Klaus Metsch

   Pages 74-85

10. Linear spaces with n2+n+1 points

    • Klaus Metsch

    Pages 86-93

11. A hypothetical structure

    • Klaus Metsch

    Pages 94-105

12. Linear spaces with n2+n+2 lines

    • Klaus Metsch

    Pages 106-117

13. Points of degree n and another characterization of the linear spaces L(n,d)

    • Klaus Metsch

    Pages 118-130

14. The non-existence of certain (7,1)-designs and determination of A(5) and A(6)

    • Klaus Metsch

    Pages 131-140

15. A result on graph theory with an application to linear spaces

    • Klaus Metsch

    Pages 141-149

16. Linear spaces in which every long line meets only few lines

    • Klaus Metsch

    Pages 150-160

17. s-fold inflated projective planes

    • Klaus Metsch

    Pages 161-180

18. The Dowling Wilson Conjecture

    • Klaus Metsch

    Pages 181-187

19. Uniqueness of embeddings

    • Klaus Metsch

    Pages 188-191

20. Back Matter

    Pages 192-196

    PDF

A famous theorem in the theory of linear spaces states that every finite linear space has at least as many lines as points. This result of De Bruijn and Erd|s led to the conjecture that every linear space with "few lines" canbe obtained from a projective plane by changing only a small part of itsstructure. Many results related to this conjecture have been proved in the last twenty years. This monograph surveys the subject and presents several new results, such as the recent proof of the Dowling-Wilsonconjecture. Typical methods used in combinatorics are developed so that the text can be understood without too much background. Thus the book will be of interest to anybody doing combinatorics and can also help other readers to learn the techniques used in this particular field.

• Combinatorics
• Embeddings
• Finite
• Linear Space
• boundary element method
• character
• design
• eXist
• field
• graph
• graph theory
• proof
• techniques
• theorem

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