Linear spaces on v points with v-2 mutually intersecting long lines
We assume familiarity with the concepts of finite linear spaces. In this paper it is proved that a finite linear space \textbf{L} such that (1) any two long lines intersect each other, (2) \(b_ n=v-2\), where \(b_ n\) is the number of long lines and v is the number of points of \textbf{L}, is one of the following structures: (i) a projective plane of order n-1 with two extra points, (ii) the Fano quasi-plane with a long line broken into three short ones, (iii) the finite linear space obtained from a projective plane of order n-1 by breaking up a line into a linear space and with an extra point, (iv) the punctured Fano plane, (v) the finite linear space obtained from a projective plane of order n-1 by breaking up two lines into two linear spaces. For the definition of these notions, and related concepts, we refer the reader also to \textit{P. de Witte} [J. Reine Angew. Math. 288, 66-73 (1976; Zbl 0333.50009)].
scientific article; zbMATH DE number 4157367