A Structural Theorem for Sets With Few Triangles

TL;DR

This paper characterizes the structure of finite point sets in the plane with minimal triangle congruence classes, showing they must have a significant collinear or circular subset, supporting Erdős conjectures.

Contribution

It provides a structural theorem linking few triangles to large collinear or circular subsets, using affine group analysis and additive combinatorics.

Findings

Sets with few triangles have many points on a line or circle.

The structure supports Erdős conjectures on geometric configurations.

Uses advanced group and combinatorics techniques.

Abstract

We show that if a finite point set $P\subseteq \mathbb{R}^2$ has the fewest congruence classes of triangles possible, up to a constant $M$, then at least one of the following holds. (1) There is a $\sigma>0$ and a line $l$ which contains $\Omega(|P|^\sigma)$ points of $P$. Further, a positive proportion of $P$ is covered by lines parallel to $l$ each containing $\Omega(|P|^\sigma)$ points of $P$. (2) There is a circle $\gamma$ which contains a positive proportion of $P$. This provides evidence for two conjectures of Erd\H{o}s. We use the result of Petridis-Roche-Newton-Rudnev-Warren on the structure of the affine group combined with classical results from additive combinatorics.