Erdős-Szekeres-Type Problems in the Real Projective Plane

Martin Balko, Manfred Scheucher, Pavel Valtr

Abstract: We consider point sets in the real projective plane \({{\,\mathrm{{\mathbb {R}}{\mathcal {P}}^2}\,}}\) 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 \({{\,\mathrm{{\mathbb {R}}{\mathcal {P}}^2}\,}}\), which was initiated by Harborth and Möller in 1994. The notion of convex position in \({{\,\mathrm{{\mathbb {R}}{\mathcal {P}}^2}\,}}\) agrees with the definition of convex sets introduced by Steinitz in 1913. For \(k \ge 3\), 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 \({{\,\mathrm{{\mathbb {R}}{\mathcal {P}}^2}\,}}\), called projective k-holes, we find arbitrarily large finite sets of points from \({{\,\mathrm{{\mathbb {R}}{\mathcal {P}}^2}\,}}\) 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 \(k \le 7\). On the other hand, we show that the number of k-holes can be substantially larger in \({{\,\mathrm{{\mathbb {R}}{\mathcal {P}}^2}\,}}\) than in \({\mathbb {R}}^2\) by constructing, for every \(k \in \{3,\dots ,6\}\), sets of n points from \({\mathbb {R}}^2 \subset {{\,\mathrm{{\mathbb {R}}{\mathcal {P}}^2}\,}}\) 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 \({{\,\mathrm{{\mathbb {R}}{\mathcal {P}}^2}\,}}\) and about some algorithmic aspects. The study of extremal problems about point sets in \({{\,\mathrm{{\mathbb {R}}{\mathcal {P}}^2}\,}}\) opens a new area of research, which we support by posing several open problems.