A Structural Theorem for Sets with Few Triangles

We show that if a finite point set P subset of R2 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 Erds. We use the result of Petridis-Roche-Newton-Rudnev-Warren on the structure of the affine group combined with classical results from additive combinatorics.