A classical open problem in combinatorial geometry is to obtain tight asymptotic bounds on the maximum number of $k$-level vertices in an arrangement of $n$ hyperplanes in ${\mathbb{R}}^d$ (vertices with exactly $k$ of the hyperplanes passing below them). This is essentially a dual version of the $k$-set problem, which, in a primal setting, seeks bounds for the maximum number of $k$-sets determined by $n$ points in ${\mathbb{R}}^d$, where a $k$-set is a subset of size $k$ that can be separated from its complement by a hyperplane. The $k$-set problem is still wide open even in the plane. In three dimensions, the best known upper and lower bounds are, respectively, $O\left(n{k}^{3∕2}\right)$ (Sharir et al., 2001) and $nk\cdot 2^{\Omega \left(\sqrt{logk}\right)}$ (Tóth, 2001).

In its dual version, the problem can be generalized by replacing hyperplanes by other families of surfaces (or curves in the planes). Reasonably sharp bounds have been obtained for curves in the plane (Sharir and Zahl, 2017 [20]; Tamaki and Tokuyama, 2003), but the known upper bounds are rather weak for more general surfaces, already in three dimensions, except for the case of triangles (Agarwal et al., 1998). The best known general bound, due to Chan (2012) [6], is $O\left(n^{2.997}\right)$, for families of surfaces that satisfy certain (fairly weak) properties.

In this paper we consider the case of pseudoplanes in ${\mathbb{R}}^3$ (defined in detail in the introduction), and establish the upper bound $O\left(n{k}^{5∕3}\right)$ for the number of $k$-level vertices in an arrangement of $n$ pseudoplanes. The bound is obtained by establishing suitable (and nontrivial) extensions of dual versions of classical tools that have been used in studying the primal $k$-set problem, such as the Lovász Lemma and the Crossing Lemma.

组合几何学中的经典开放问题是获得最大个数上的紧渐近界 $ķ$-以以下方式排列的水平顶点$ñ$ 超飞机 ${\mathbb{[R}}^d$ （顶点与 $ķ$在它们下面通过的超平面的数量）。这本质上是$ķ$-问题集，在原始情况下，它寻求最大数量的界限$ķ$-套由下式确定$ñ$ 点入 ${\mathbb{[R}}^d$，其中 $ķ$-set是大小的子集 $ķ$可以通过超平面将其与其补体分开。这$ķ$设置问题即使在飞机上也仍然敞开。在三个维度上，最知名的上限和下限分别是$Ø（ñ{ķ}^{3∕\mathrm{2个}}）$ （Sharir et al。，2001）和 $ñķ\cdot {\mathrm{2个}}^{\Omega （\sqrt{日志ķ}）}$ （2001年，托特）。

在其双重版本中，可以通过用其他系列的曲面（或平面中的曲线）替换超平面来推广该问题。在平面上的曲线上已经获得了合理的尖锐边界（Sharir和Zahl，2017 [20]； Tamaki和Tokuyama，2003），但是对于更一般的曲面，已知的上限已经很弱了，除了三维外，已经存在三个维度三角形（Agarwal et al。，1998）。由于Chan（2012）[6]，最著名的一般界是$Ø（{ñ}^{\mathrm{2个}。997}）$，用于满足某些（相当弱）特性的表面族。

在本文中，我们考虑的情况下pseudoplanes在${\mathbb{[R}}^3$ （在引言中详细定义），并确定上限 $Ø（ñ{ķ}^{5∕3}）$ 对于的数量 $ķ$顶点的排列 $ñ$伪平面。通过建立用于研究原始模型的经典工具的双重版本的适当（且不平凡的）扩展来获得边界$ķ$问题，例如Lovász引理和Crossing Lemma。