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 part of the set has a structure. 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. [ABSTRACT FROM AUTHOR]