Linear spaces with a family intersecting lines (Q1866756)

Recall that the order of a linear space \(S\) is \(n\) iff \(n+1\) is the maximal number of lines passing through any point. A clique is a set of lines which mutually intersect. The author proves the following theorem: If \(b_0\geq n^2-n+2\) is the size of a maximal clique in \(S\) then \(S\) is a projective plane, a \(1\)-punctured, or a \(2\)-punctured projective plane. Therefore \(b_0=n^2+n+1, n^2+1, n^2-n+2,\) respectively. Some statements for the case \(b_0=n^2-n+1\) are given as well.