Maximum number of almost similar triangles in the plane

József Balogh, Felix Christian Clemen, Bernard Lidický

Research output: Contribution to journalArticlepeer-review


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)=(1+o(1))n3/24. Exploring connections to hypergraph Turán problems, we use flag algebras and stability techniques for the proof.

Original languageEnglish (US)
Article number101880
JournalComputational Geometry: Theory and Applications
StatePublished - Aug 1 2022


  • Extremal hypergraphs
  • Flag algebras
  • Similar triangles

ASJC Scopus subject areas

  • Computer Science Applications
  • Geometry and Topology
  • Control and Optimization
  • Computational Theory and Mathematics
  • Computational Mathematics


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