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arXiv · 2402.07775

Growth Rate of the Number of Empty Triangles in the Plane

Bhaswar B. Bhattacharya·Sandip Das·Sk Samim Islam·Saumya Sen
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Abstract

Given a set $P$ of $n$ points in the plane, in general position, denote by $N_\Delta(P)$ the number of empty triangles with vertices in $P$. In this paper we investigate by how much $N_\Delta(P)$ changes if a point $x$ is removed from $P$. By constructing a graph $G_P(x)$ 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 $G_P(x)$. We study properties of the graph $G_P(x)$ and, in particular, show that it is kite-free. This relates the growth rate of the number of empty triangles to the famous Ruzsa-Szemer\'edi problem.

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Bhaswar B. Bhattacharya, Sandip Das, Sk Samim Islam, Saumya Sen. 2024-02-12. Growth Rate of the Number of Empty Triangles in the Plane. https://arxiv.org/abs/2402.07775

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