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arXiv · 2101.10304

Maximum Number of Almost Similar Triangles in the Plane

József Balogh·Felix Christian Clemen·Bernard Lidický
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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.

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József Balogh, Felix Christian Clemen, Bernard Lidický. 2021-01-25. Maximum Number of Almost Similar Triangles in the Plane. https://doi.org/10.1016/j.comgeo.2022.101880

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