Erd\H{o}s--Szekeres-type problems in the real projective plane

TL;DR

This paper investigates extremal problems for point sets in the real projective plane, providing bounds on convex configurations and introducing new notions of holes, revealing differences from Euclidean cases and opening new research directions.

Contribution

It introduces the concept of projective k-holes, establishes bounds on their numbers, and compares their properties to Euclidean cases, extending classical extremal geometry to the projective setting.

Findings

Asymptotically tight bounds for convex position in $ ext{RP}^2$

Existence of large point sets with no projective 8-holes

Quadratic bounds on the number of projective k-holes for k ≤ 7

Abstract

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 Erd\H{o}s--Szekeres-type problems. We provide asymptotically tight bounds for a variant of the Erd\H{o}s--Szekeres theorem about point sets in convex position in $\mathbb{R}P^2$, which was initiated by Harborth and M\"oller 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 $\Omega(n^{3-3/5k})$ \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.