LINEAR-SPACES AND PARTITIONING THE PROJECTIVE PLANE

The aim of this paper is to settle a question about the partitioning o f the projective plane by lines except for a small set. Suppose that Q is a set of points in the projective plane of order n and Pi is a set of lines that partitions the complement of Q. If Q has at most 2n - 1 points and P has less than n + 1 + root n lines, then these lines are concurrent. An example is given which shows that the condition on the number of points of Q is sharp. However, it turns out that this is a 'pathological' example and if we exclude this case, then the statement can be improved. (C) 1997 Academic Press.