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Full Description
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.
Contents
Definition and basic properties of linear spaces.- Lower bounds for the number of lines.- Basic properties and results of (n+1,1)-designs.- Points of degree n.- Linear spaces with few lines.- Embedding (n+1,1)-designs into projective planes.- An optimal bound for embedding linear spaces into projective planes.- The theorem of totten.- Linear spaces with n2+n+1 points.- A hypothetical structure.- Linear spaces with n2+n+2 lines.- Points of degree n and another characterization of the linear spaces L(n,d).- The non-existence of certain (7,1)-designs and determination of A(5) and A(6).- A result on graph theory with an application to linear spaces.- Linear spaces in which every long line meets only few lines.- s-fold inflated projective planes.- The Dowling Wilson Conjecture.- Uniqueness of embeddings.
