Linear Spaces and Partitioning the Projective Plane

The aim of this paper is to settle a question about the partitioning of the projective plane by lines except for a small set. Suppose thatQis a set of points in the projective plane of ordernandΠis a set of lines that partitions the complement ofQ. IfQhas at most 2n−1 points andPhas less thann+1+nlines, then these lines are concurrent. An example is given which shows that the condition on the number of points ofQis sharp. However, it turns out that this is a ‘pathological’ example and if we exclude this case, then the statement can be improved.