The minimal number of lines intersected by a set of \(q+2\) points, blocking sets, and intersecting circles (Q1824202)

The authors give some lower bounds for the number of lines intersecting a set of \(q+2\) points in a projective plane of order q. As an application one deduces some upper bounds for the number of circles double self intersecting in a Möbius plane of order n. In particular Lemma 1.1, Theorems 1.2, 1.3, 1.4, 1.7, 3.1, 3.2 should be pointed out. For the sake of shortness we recall here only Lemma 1.1: Let S be a set of \(q+2\) points in a projective plane \(\pi\) of order q and A be a subset of S. Assume that for any point P of S-A the number of tangents at A passing through P is not bigger than \(| A| -\lambda,\) where \(\lambda\) is an integer \(\geq 1\). Than the number of lines in \(\pi\) intersecting S is at least \(\binom{q+2}{2}+(\lambda | 2)| S-A|.\)