Some linear spaces with lines of two lengths (Q1262527)

Let (P,L) be a finite linear space with exactly m or n points on each line, n-m\(\geq 2\), \(g.c.d(m-1,n-m)=1\), and p the difference of the maximal number r of lines through a point of S and the minimal numbers s. Furthermore, let p be a prime. Properties of such linear spaces are investigated. Examples are degenerate projective planes, complete graphs of \(| P| -\ell \cdot n\) vertices, together with \(\ell\) mutually skew n-lines, and Steiner systems S(2,4,v) in which one block is broken into pairs of points. In each of these examples one has \(m=2\).