Finite linear spaces with \(b-v \in \{M,M+1\}\) (Q1849882)

An interesting problem which has been intensively investigated in finite geometry is to establish how the fundamental parameters \(v\) and \(b\) of a finite linear space are related. A famous result due to de Bruijn and Erdős going back to 1948 states that \(b\geq v\). Given a positive integer \(s\), the classification of finite linear spaces with \(b - v\leq s\) is the major problem of current interest. The case \(s =\sqrt{v}\) was settled by Totten in 1976 and Totten's result was refined by Metsch in 1991. Let \(M + 1\) be the maximum line length, and let \(n + 1\) denote the maximum number of lines on a point (the order of the linear space). In the paper under review the author considers the problem of classifying finite linear spaces with \(b - v \in\{M, M+ 1\}\). He gives a complete classification for the case \(b - v = M\), and \(b - v = M + 1\neq n\). Finally, some results on the case \(b - v = M + 1 = n\) are shown. He describes four types of such finite linear spaces, and shows that either the linear space is one of these four types, or there are two lines of length \(M + 1 = n\) and \(b\geq n^2 - n + 3\). Moreover, if \(b = n^2 - n + 3\), then \(n\leq 10\).