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.

• 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.

• There is a circle \(\gamma \) which contains a positive proportion of P.

This provides evidence for two conjectures of Erdős. We use the result of Petridis–Roche–Newton–Rudnev–Warren on the structure of the affine group combined with classical results from additive combinatorics.

我们证明，如果有限点集\(P\subseteq {\mathbb {R}}^2\)具有尽可能少的三角形同余类，最多为常数M，则以下至少之一成立。

• 有一条\(\sigma >0\)和一条包含P的\(\Omega (|P|^\sigma )\)个点的线l。此外， P的正比例被平行于l的线覆盖，每条线包含P的\(\Omega (|P|^\sigma )\)个点。

• 有一个圆\(\gamma \) ，其中包含P的正比例。

这为埃尔多斯的两个猜想提供了证据。我们使用 Petridis-Roche-Newton-Rudnev-Warren 关于仿射群结构的结果以及加性组合学的经典结果。