Abstract. Let P be a set of n points in the plane, not all on a line. We show that if n is large then there are at least n/2 ordinary lines, that is to say lines passing through exactly two points of P . This confirms, for large n, a conjecture of Dirac and Motzkin. In fact we describe the exact extremisers for this problem, as well as all sets having fewer than n − C ordinary lines for some absolute constant C. We also solve, for large n, the "orchard-planting problem", which asks for the maximum number of lines through exactly 3 points of P . Underlying these results is a structure theorem which states that if P has at most Kn ordinary lines then all but O(K) points of P lie on a cubic curve, if n is sufficiently large depending on K.

No abstract