Erdős–Szekeres-Type Problems in the Real Projective Plane

We consider point sets in the real projective plane RP² 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 RP², which was initiated by Harborth and Möller in 1994. The notion of convex position in RP² agrees with the definition of convex sets introduced by Steinitz in 1913. For k≥3, an (affine) k-hole in a finite set S⊆R² 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 RP², called projective k-holes, we find arbitrarily large finite sets of points from RP² 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≤7. On the other hand, we show that the number of k-holes can be substantially larger in RP² than in R² by constructing, for every k∈{3,⋯,6}, sets of n points from R²⊂RP² with Ω(n^3-3/5k) projective k-holes and only O(n²) affine k-holes. Last but not least, we prove several other results, for example about projective holes in random point sets in RP² and about some algorithmic aspects. The study of extremal problems about point sets in RP² opens a new area of research, which we support by posing several open problems.