Maximum Number of Almost Similar Triangles in the Plane

Abstract

A triangle $T^{\prime }$ is $\epsilon$-similar to another triangle $T$ if their angles pairwise differ by at most $\epsilon$. Given a triangle $T$, $\epsilon >0$ and $n\in \mathbb{N}$, Bárány and Füredi asked to determine the maximum number of triangles $h(n,T,\epsilon )$ being $\epsilon$-similar to $T$ in a planar point set of size $n$. We show that for almost all triangles $T$ there exists $\epsilon =\epsilon (T)>0$ such that $h(n,T,\epsilon )=n^3/24(1+o(1))$. Exploring connections to hypergraph Turán problems, we use flag algebras and stability techniques for the proof.

20 pages

References in corpus (3)

• Minimum number of monotone subsequences of length 4 in permutations
• Improving bounds on packing densities of 4-point permutations
• Closing in on Hill's conjecture