A remark on partial linear spaces of girth 5 with an application to strongly regular graphs (Q1117938)

A partial linear space is an incidence structure (P,L,I), where the line- set L is the set of subsets of the point-set P and where any two lines have at most one point in common. Collinear points are called neighbours. The girth of a partial linear space is the length of a shortest circuit. Let k be a number of points of the point-set P. The authors prove the following theorem. Theorem. A connected partial linear space with girth at least 5 and more than one line (lines possibly of varying size) in which every point has \(\lambda\) neighbours, contains \(k\geq \lambda (\lambda +3)/2\) points. In view of the application to strongly regular graphs with parameters v, k, \(\lambda\), \(\mu\) there is observed the following corollary. Corollary. A strongly regular graph with \(\mu =2\) and \(k<\lambda (\lambda +3)/2\) is a partial quadrangle; in particular it satisfies the divisibility condition \((\lambda +1)/k.\)