Maximum Number of Almost Similar Triangles in the Plane

TL;DR

This paper investigates the maximum number of triangles similar to a given triangle within a set of points, establishing asymptotic bounds for almost all triangles using advanced combinatorial methods.

Contribution

It proves that for almost all triangles, there exists a positive epsilon such that the maximum number of epsilon-similar triangles in a point set of size n is asymptotically n^3/24, employing flag algebras and stability techniques.

Findings

Maximum number of epsilon-similar triangles is asymptotically n^3/24 for almost all triangles.

Uses flag algebra and stability methods from hypergraph Turán problems.

Establishes existence of epsilon depending on the triangle T.

Abstract

A triangle $T'$ is $\varepsilon$-similar to another triangle $T$ if their angles pairwise differ by at most $\varepsilon$. Given a triangle $T$, $\varepsilon>0$ and $n\in\mathbb{N}$, B\'ar\'any and F\"uredi asked to determine the maximum number of triangles $h(n,T,\varepsilon)$ being $\varepsilon$-similar to $T$ in a planar point set of size $n$. We show that for almost all triangles $T$ there exists $\varepsilon=\varepsilon(T)>0$ such that $h(n,T,\varepsilon)=n^3/24 (1+o(1))$. Exploring connections to hypergraph Tur\'an problems, we use flag algebras and stability techniques for the proof.