THERE EXIST 6N/13 ORDINARY POINTS

In 1958 L. M. Kelly and W. O. J. Moser showed that apart from a pencil, any configuration of n lines in the real projective plane has at least 3n/7 ordinary or simple points of intersection, with equality in the Kelly Moser example (a complete quadrilateral with its three diagonal lines). In 1981 S. Hansen claimed to have improved this to n/2 (apart from pencils, the Kelly-Moser example and the McKee example). In this paper we show that one of the main theorems used by Hansen is false, thus leaving n/2 open, and we improve the 3n/7 estimate to 6n/13 (apart from pencils and the Kelly Moser example), with equality in the McKee example. Our result applies also to arrangements of pseudolines.