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