Erdős–Szekeres-type problems in the real projective plane
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We consider point sets in the real projective plane ℝP^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 ℝP^2, which was initiated by Harborth and Möller in 1994. The notion of convex position in ℝP^2 agrees with the definition of convex sets introduced by Steinitz in 1913. For k ≥ 3, an () k-hole in a finite set S ⊆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 ℝP^2, called projective k-holes, we find arbitrarily large finite sets of points from ℝP^2 with no 8-holes, providing an analogue of a classical planar construction by Horton from 1983. We also prove that they contain only quadratically many k-holes for k ≤ 7. On the other hand, we show that the number of k-holes can be substantially larger in ℝP^2 than in ℝ^2 by constructing, for every k ∈{3,…,6}, sets of n points from ℝ^2 ⊂ℝP^2 with Ω(n^3-3/5k) k-holes and only O(n^2) k-holes. Last but not least, we prove several other results, for example about projective holes in random point sets in ℝP^2 and about some algorithmic aspects. The study of extremal problems about point sets in ℝP^2 opens a new area of research, which we support by posing several open problems.
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