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.

中文翻译：

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几乎全等的三角形

摘要

大约 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于等边三角形？他们指出，通过将点均匀地分成围绕固定等边三角形的三个顶点构建的三个小簇，至少可以得到\(\left\lfloor \frac{n}{3} \right\rfloor \cdot \left \lfloor \frac{n+1}{3} \right\rfloor \cdot \left\lfloor \frac{n+2}{3} \right\rfloor \)这样的近似副本。在本文中，我们提供了一个匹配的上限，从而回答了他们的问题。更一般地，对于每个三角形T ，我们确定大小为n的点集中与T近似全等三角形的最大数量。我们的部分证明基于超图图兰理论：对于平面上的每个点集和三角形T，我们构造一个 3-均匀超图\(\mathcal {H}=\mathcal {H}(T)\)，其中不包含超图作为来自禁止超图族的子图\(\mathcal {F}=\mathcal {F}(T)\)。我们对\(\mathcal {H}\)边数的上限将确定与T近似全等的三角形的最大数量。