Linear Spaces with Few Lines

Klaus Metsch (auth.)3540547207, 9783540547204, 0387547207

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.

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Table of contents :
Definition and basic properties of linear spaces….Pages 1-8
Lower bounds for the number of lines….Pages 9-14
Basic properties and results of (n+1,1)-designs….Pages 15-20
Points of degree n….Pages 21-30
Linear spaces with few lines….Pages 31-42
Embedding (n+1,1)-designs into projective planes….Pages 43-60
An optimal bound for embedding linear spaces into projective planes….Pages 61-73
The theorem of totten….Pages 74-85
Linear spaces with n 2 +n+1 points….Pages 86-93
A hypothetical structure….Pages 94-105
Linear spaces with n 2 +n+2 lines….Pages 106-117
Points of degree n and another characterization of the linear spaces L(n,d)….Pages 118-130
The non-existence of certain (7,1)-designs and determination of A(5) and A(6)….Pages 131-140
A result on graph theory with an application to linear spaces….Pages 141-149
Linear spaces in which every long line meets only few lines….Pages 150-160
s-fold inflated projective planes….Pages 161-180
The Dowling Wilson Conjecture….Pages 181-187
Uniqueness of embeddings….Pages 188-191