{"title":"Almost Congruent Triangles","authors":"","doi":"10.1007/s00454-023-00623-9","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>Almost 50 years ago Erdős and Purdy asked the following question: Given <em>n</em> 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 <span> <span>\\(\\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 \\)</span> </span> such approximate copies. In this paper we provide a matching upper bound and thereby answer their question. More generally, for every triangle <em>T</em> we determine the maximum number of approximate congruent triangles to <em>T</em> in a point set of size <em>n</em>. Parts of our proof are based on hypergraph Turán theory: for each point set in the plane and a triangle <em>T</em>, we construct a 3-uniform hypergraph <span> <span>\\(\\mathcal {H}=\\mathcal {H}(T)\\)</span> </span>, which contains no hypergraph as a subgraph from a family of forbidden hypergraphs <span> <span>\\(\\mathcal {F}=\\mathcal {F}(T)\\)</span> </span>. Our upper bound on the number of edges of <span> <span>\\(\\mathcal {H}\\)</span> </span> will determine the maximum number of triangles that are approximate congruent to <em>T</em>.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"75 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete & Computational Geometry","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00454-023-00623-9","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
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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.
期刊介绍：
Discrete & Computational Geometry (DCG) is an international journal of mathematics and computer science, covering a broad range of topics in which geometry plays a fundamental role. It publishes papers on such topics as configurations and arrangements, spatial subdivision, packing, covering, and tiling, geometric complexity, polytopes, point location, geometric probability, geometric range searching, combinatorial and computational topology, probabilistic techniques in computational geometry, geometric graphs, geometry of numbers, and motion planning.
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