József Solymosi: On the structure of pointsets with many collinear triples

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 will show that under some combinatorial conditions, such pointsets have 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 point set in a convex position is small, then many points are on a conic.