On the structure of pointsets with many collinear triples

TL;DR

This paper investigates the structure of point sets with many collinear triples, proving a case of Elekes' conjecture and establishing a density version of Jamison's theorem related to points on conics.

Contribution

It demonstrates that point sets with many collinear triples contain special configurations, confirming part of Elekes' conjecture and linking small directional diversity to conic structures.

Findings

Point sets with many collinear triples have specific configurations.

A density version of Jamison's theorem is established.

Small number of directions implies many points lie on a conic.

Abstract

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.