Linear spaces with a regularity condition (Q1199976)

A linear space is an incidence structure such that any line contains at least two points and through any two different points exactly one line is passing. In the paper finite linear spaces with the following regularity property are studied: There exist two integers \(s,t\) such that for any two disjoint lines \(p,q\) and point \(P\) outside of them are either \(s\) or \(t\) lines going through \(P\) and intersecting \(p\) and \(q\). The case \(s=t\) was described completely in an earlier common paper of A. Delandtsheer and the second author. For \(s\neq t\) the situation looks more complicated and it is very difficult to classify all such linear spaces. In the paper several properties of linear spaces with regularity \(s\neq t\) are presented and two special classes are described.