TY - JOUR

T1 - Triangle areas in line arrangements

AU - Damásdi, Gábor

AU - Martínez-Sandoval, Leonardo

AU - Nagy, Dániel T.

AU - Nagy, Zoltán Lóránt

N1 - Funding Information:
Supported by the ÚNKP-18-3 New National Excellence Program of the Ministry of Human Capacities.Supported by the grant ANR-17-CE40-0018 of the French National Research Agency ANR (project CAPPS).Research supported by National Research, Development and Innovation Office - NKFIH grants K 116769, K 132696 and FK 132060.Supported by the Hungarian Research Grant (OTKA)No. K 120154 and by the János Bolyai Scholarship of the Hungarian Academy of Sciences.
Publisher Copyright:
© 2020 The Author(s)

PY - 2020/12/1

Y1 - 2020/12/1

N2 - A widely investigated subject in combinatorial geometry, originated from Erdős, is the following. Given a point set P of cardinality n in the plane, how can we describe the distribution of the determined distances? This has been generalized in many directions. In this paper we propose the following variants. What is the maximum number of triangles of unit area, maximum area or minimum area, that can be determined by an arrangement of n lines in the plane? We prove that the order of magnitude for the maximum occurrence of unit areas lies between Ω(n2) and O(n9∕4+ε), for every ε>0. This result is strongly connected to additive combinatorial results and Szemerédi–Trotter type incidence theorems. Next we show an almost tight bound for the maximum number of minimum area triangles. Finally, we present lower and upper bounds for the maximum area and distinct area problems by combining algebraic, geometric and combinatorial techniques.

AB - A widely investigated subject in combinatorial geometry, originated from Erdős, is the following. Given a point set P of cardinality n in the plane, how can we describe the distribution of the determined distances? This has been generalized in many directions. In this paper we propose the following variants. What is the maximum number of triangles of unit area, maximum area or minimum area, that can be determined by an arrangement of n lines in the plane? We prove that the order of magnitude for the maximum occurrence of unit areas lies between Ω(n2) and O(n9∕4+ε), for every ε>0. This result is strongly connected to additive combinatorial results and Szemerédi–Trotter type incidence theorems. Next we show an almost tight bound for the maximum number of minimum area triangles. Finally, we present lower and upper bounds for the maximum area and distinct area problems by combining algebraic, geometric and combinatorial techniques.

KW - Algebraic curves

KW - Extremal problems

KW - Incidence theorems

KW - Unit area triangles

UR - http://www.scopus.com/inward/record.url?scp=85089481721&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2020.112105

DO - 10.1016/j.disc.2020.112105

M3 - Article

AN - SCOPUS:85089481721

VL - 343

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 12

M1 - 112105

ER -