Growth Rate of the Number of Empty Triangles in the Plane

Document Type

Conference Article

Publication Title

Lecture Notes in Computer Science Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics

Abstract

Given a set P of n points in the plane, in general position, denote by NΔ(P) the number of empty triangles with vertices in P. In this paper we investigate by how much NΔ(P) changes if a point x is removed from P. By constructing a graph GP(x) based on the arrangement of the empty triangles incident on x, we transform this geometric problem to the problem of counting triangles in the graph GP(x). We study properties of the graph GP(x) and, in particular, show that it is kite-free. This relates the growth rate of the number of empty triangles to the famous Ruzsa-Szemerédi problem.

First Page

77

Last Page

87

DOI

10.1007/978-3-031-52213-0_6

Publication Date

1-1-2024

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