Year: 2023
Type: article
Abstract: We show that if a finite point set $$P\subseteq {\mathbb {R}}^2$$ P ⊆ 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 $$\sigma >0$$ σ > 0 and a line l which contains $$\Omega (|P|^\sigma )$$ Ω ( | P | σ ) points of P. Further, a positive proportion of P is covered by lines parallel to l each containing $$\Omega (|P|^\sigma )$$ Ω ( | P | σ ) points of P. There is a circle $$\gamma $$ γ 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.
Source: COMBINATORICA
Authors: Sam Mansfield, Jonathan Passant
Institution: University of Bristol
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
Award: G7192499587
OpenAlex ID: W4387571769