TY - JOUR

T1 - Almost Congruent Triangles

AU - Balogh, József

AU - Clemen, Felix Christian

AU - Dumitrescu, Adrian

N1 - Publisher Copyright:
© 2024, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2024

Y1 - 2024

N2 - 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 ⌊n3⌋·⌊n+13⌋·⌊n+23⌋ 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 H= H(T) , which contains no hypergraph as a subgraph from a family of forbidden hypergraphs F= F(T) . Our upper bound on the number of edges of H will determine the maximum number of triangles that are approximate congruent to T.

AB - 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 ⌊n3⌋·⌊n+13⌋·⌊n+23⌋ 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 H= H(T) , which contains no hypergraph as a subgraph from a family of forbidden hypergraphs F= F(T) . Our upper bound on the number of edges of H will determine the maximum number of triangles that are approximate congruent to T.

KW - Congruent triangles

KW - Hypergraphs

KW - Lagrangian method

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U2 - 10.1007/s00454-023-00623-9

DO - 10.1007/s00454-023-00623-9

M3 - Article

AN - SCOPUS:85182207909

SN - 0179-5376

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

ER -