Maximum Number of Almost Similar Triangles in the Plane

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.