Almost Congruent Triangles

Pub Date : 2024-01-11 DOI:10.1007/s00454-023-00623-9

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Abstract

Almost 50 years ago Erdős and Purdy asked the following question: Given n points in the plane, how many triangles can be approximate congruent to equilateral triangles? They pointed out that by dividing the points evenly into three small clusters built around the three vertices of a fixed equilateral triangle, one gets at least \(\left\lfloor \frac{n}{3} \right\rfloor \cdot \left\lfloor \frac{n+1}{3} \right\rfloor \cdot \left\lfloor \frac{n+2}{3} \right\rfloor \) such approximate copies. In this paper we provide a matching upper bound and thereby answer their question. More generally, for every triangle T we determine the maximum number of approximate congruent triangles to T in a point set of size n. Parts of our proof are based on hypergraph Turán theory: for each point set in the plane and a triangle T, we construct a 3-uniform hypergraph \(\mathcal {H}=\mathcal {H}(T)\) , which contains no hypergraph as a subgraph from a family of forbidden hypergraphs \(\mathcal {F}=\mathcal {F}(T)\) . Our upper bound on the number of edges of \(\mathcal {H}\) will determine the maximum number of triangles that are approximate congruent to T.

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几乎全等的三角形

摘要将近 50 年前，Erdős 和 Purdy 提出了以下问题：给定平面上的 n 个点，有多少个三角形可以近似全等？他们指出，通过把点平均分成围绕固定等边三角形三个顶点的三个小群，至少可以得到（\left\lfloor \frac{n}{3}\cdot \left\lfloor \frac{n+1}{3}\（rightrfloor）（cdot）（leftlfloor）（frac{n+2}{3}\这样的近似副本。在本文中，我们提供了一个匹配的上界，从而回答了他们的问题。更广义地说，对于每个三角形 T，我们都要确定在大小为 n 的点集中与 T 近似全等的三角形的最大数目。我们的证明部分基于超图图兰理论：对于平面中的每个点集和三角形 T，我们都要构造一个 3-Uniform 超图 \(\mathcal {H}=\mathcal {H}(T)\) ，其中不包含任何超图，因为它是一个全等的三角形。(\mathcal{F}=\mathcal{F}(T)\))的子图中不包含任何超图。我们对 \(\mathcal {H}\) 边缘数量的上限将决定与 T 近似全等的三角形的最大数量。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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