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.

翻译：给定一般位置上的平面点集 $P$（包含 $n$ 个点），记 $N_\Delta(P)$ 为顶点取自 $P$ 的空三角形数量。本文研究当从 $P$ 中移除一个点 $x$ 时，$N_\Delta(P)$ 的变化量。通过基于与 $x$ 相邻的空三角形排列构造图 $G_P(x)$，我们将该几何问题转化为对图 $G_P(x)$ 中三角形计数的问题。我们研究了图 $G_P(x)$ 的性质，特别证明了其无风筝图结构。这一发现将空三角形数量的增长率与著名的 Ruzsa–Szemerédi 问题联系起来。