TY - JOUR
T1 - Maximum number of almost similar triangles in the plane
AU - Balogh, József
AU - Clemen, Felix Christian
AU - Lidický, Bernard
N1 - Research is partially supported by NSF Grant DMS-1764123, NSF RTG grant DMS 1937241, Arnold O. Beckman Research Award (UIUC Campus Research Board RB 18132), the Langan Scholar Fund (UIUC), and the Simons Fellowship.Research of this author is partially supported by NSF grant DMS-1855653.
PY - 2022/8/1
Y1 - 2022/8/1
N2 - 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.
AB - 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.
KW - Extremal hypergraphs
KW - Flag algebras
KW - Similar triangles
UR - http://www.scopus.com/inward/record.url?scp=85127500987&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85127500987&partnerID=8YFLogxK
U2 - 10.1016/j.comgeo.2022.101880
DO - 10.1016/j.comgeo.2022.101880
M3 - Article
AN - SCOPUS:85127500987
SN - 0925-7721
VL - 105-106
JO - Computational Geometry: Theory and Applications
JF - Computational Geometry: Theory and Applications
M1 - 101880
ER -