  It is conjectured that if a finite set of points in the plane contains many
collinear triples then there is some structure in the set. We are going to show
that under some combinatorial conditions such pointsets contain special
configurations of triples, proving a case of Elekes' conjecture. Using the
techniques applied in the proof we show a density version of Jamison's theorem.
If the number of distinct directions between many pairs of points of a pointset
in convex position is small, then many points are on a conic.
