Growth Rate of the Number of Empty Triangles in the Plane
<|reference_start|>Growth Rate of the Number of Empty Triangles in the Plane: 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.<|reference_end|>
@article{bhattacharya2024growth,
    title={Growth Rate of the Number of Empty Triangles in the Plane},
    author={Bhaswar B. Bhattacharya, Sandip Das, Sk Samim Islam, Saumya Sen},
    journal={arXiv preprint arXiv:2402.07775},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.07775},
    primaryClass={cs.DM}
}