Linear spaces with few lines / Klaus Metsch

A finite linear space is an incidence structure of points and lines, where any two points are on exactly one line, every line has at least two points, and there are at least two lines.
The well-known result of de Bruijn and Erdős (1948) states that a linear space has at least as many lines as points, with equality only if it is a projective plane or a near-pencil. This result led to the conjecture that most of those linear spaces which have few lines (that is, almost the same number of lines as points) can be obtained from projective planes by changing only a small part of their structure. Since the early 1970s several results have been obtained on the topic of this embedding problem. In the past few years some of them have been improved, and essential new results have been attained by the author. Both these new results, and the most important old ones (several times with new proofs), occur in this publication. ... (MathSciNet)