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- W4221147867 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 ErdH{o}s--Szekeres-type problems. We provide asymptotically tight bounds for a variant of the ErdH{o}s--Szekeres theorem about point sets in convex position in $mathbb{R}P^2$, which was initiated by Harborth and Moller 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." @default.
- W4221147867 created "2022-04-03" @default.
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- W4221147867 date "2022-03-14" @default.
- W4221147867 modified "2023-09-26" @default.
- W4221147867 title "ErdH{o}s--Szekeres-type problems in the real projective plane" @default.
- W4221147867 doi "https://doi.org/10.48550/arxiv.2203.07518" @default.
- W4221147867 hasPublicationYear "2022" @default.
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