Journal or Book Title
A triangle T′ is ε-similar to another triangle T if their angles pairwise differ by at most ε. Given a triangle T, ε>0 and n∈N, Bárány and Füredi asked to determine the maximum number of triangles h(n,T,ε) being ε-similar to T in a planar point set of size n. We show that for almost all triangles T there exists ε=ε(T)>0 such that h(n,T,ε)=n3/24(1+o(1)). Exploring connections to hypergraph Turán problems, we use flag algebras and stability techniques for the proof.
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Balogh, József; Clemen, Felix Christian; and Lidicky, Bernard, "Maximum Number of Almost Similar Triangles in the Plane" (2021). Mathematics Publications. 264.