On Sets Defining Few Ordinary Lines

Let $$P$$P be a set of $$n$$n points in the plane, not all on a line. We show that if $$n$$n is large then there are at least $$n/2$$n/2ordinary lines, that is to say lines passing through exactly two points of $$P$$P. This confirms, for large $$n$$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 $$n-C$$n-C ordinary lines for some absolute constant $$C$$C. We also solve, for large $$n$$n, the “orchard-planting problem”, which asks for the maximum number of lines through exactly 3 points of $$P$$P. Underlying these results is a structure theorem which states that if $$P$$P has at most $$Kn$$Kn ordinary lines then all but O(K) points of $$P$$P lie on a cubic curve, if $$n$$n is sufficiently large depending on $$K$$K.