arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2203.07518math.COcs.CG
收藏

实射影平面上的Erdens-Szekeres型问题

Erdős--Szekeres-type problems in the real projective plane

Martin Balko, Manfred Scheucher, Pavel Valtr

备注：
The extended abstract appeared at the 38th International Symposium on Computational Geometry (SoCG 2022)
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

我们考虑点集在真正的射影平面$\mathbb{R}P^2$和探讨经典极值问题的变种在此设置的平面点集，主要集中在Erdens-Szekeres型问题。我们提供了一个变种的Erdens-Szekeres定理关于点集在凸位置的$\mathbb{R}P^2$，这是由Harborth和Möller在1994年发起的。$\mathbb{R}P^2$中的凸位置概念与Steinitz在1913年引入的凸集定义一致。对于$k \geq 3$，有限集合$S \subseteq \mathbb{R}^2$中的（\affine）$k$ -洞是来自$S$的处于凸位置的$k$点的集合，并且在它们的凸包内部没有$S$的点。在介绍了一个新的概念$k$ -洞的点集从$\mathbb{R}P^2$，称为投影$k$ -洞，我们发现任意大的有限点集从$\mathbb{R}P^2$没有\projective 8-洞，提供了一个类似的经典平面结构霍顿从1983年。我们还证明了它们只包含二次方多个\projective $k$ -洞的$k \leq 7$。另一方面，我们表明，$k$ -孔的数量可以大大大于在$\mathbb{R}P^2$比在$\mathbb{R}^2$通过构建，为每个$k \in \{3,\dots,6\}$，集$n$点从$\mathbb{R}^2 \subset \mathbb{R}P^2$与$Ω(n^{3-3/5k})$ \projective \projective $k$ -孔和只有$O(n^2)$ \affine $k$ -孔。最后，我们证明了其他几个结果，例如关于$\mathbb{R}P^2$中随机点集上的投影洞和一些算法方面的结果。$\mathbb{R}P^2$中关于点集极值问题的研究开辟了一个新的研究领域，我们通过提出几个开放问题来支持这一研究。

英文摘要：

We consider point sets in the real projective plane $\mathbb{R}P^2$ and explore variants of classical extremal problems about planar point sets in this setting, with a main focus on Erdős--Szekeres-type problems. We provide asymptotically tight bounds for a variant of the Erdős--Szekeres theorem about point sets in convex position in $\mathbb{R}P^2$, which was initiated by Harborth and Möller in 1994. The notion of convex position in $\mathbb{R}P^2$ agrees with the definition of convex sets introduced by Steinitz in 1913. For $k \geq 3$, an (\affine) $k$-hole in a finite set $S \subseteq \mathbb{R}^2$ is a set of $k$ points from $S$ in convex position with no point of $S$ in the interior of their convex hull. After introducing a new notion of $k$-holes for points sets from $\mathbb{R}P^2$, called projective $k$-holes, we find arbitrarily large finite sets of points from $\mathbb{R}P^2$ with no \projective 8-holes, providing an analogue of a classical planar construction by Horton from 1983. We also prove that they contain only quadratically many \projective $k$-holes for $k \leq 7$. On the other hand, we show that the number of $k$-holes can be substantially larger in~$\mathbb{R}P^2$ than in $\mathbb{R}^2$ by constructing, for every $k \in \{3,\dots,6\}$, sets of $n$ points from $\mathbb{R}^2 \subset \mathbb{R}P^2$ with $Ω(n^{3-3/5k})$ \projective $k$-holes and only $O(n^2)$ \affine $k$-holes. Last but not least, we prove several other results, for example about projective holes in random point sets in $\mathbb{R}P^2$ and about some algorithmic aspects. The study of extremal problems about point sets in $\mathbb{R}P^2$ opens a new area of research, which we support by posing several open problems.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑