Manfred Scheucher: Erdős-Szekeres-type problems in the real projective plane

In this talk we consider point sets in the real projective plane $R{P}^2$ 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 $R{P}^2$, which was initiated by Harborth and M\"oller in 1994. The notion of convex position in $R{P}^2$ agrees with the definition of convex sets introduced by Steinitz in 1913. An affine $k$-hole in a finite set $S\subseteq R^2$ 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 $R{P}^2$, called projective $k$-holes, we find arbitrarily large finite sets of points from $R{P}^2$ with no projective 8-holes, providing an analogue of a classical result by Horton from 1983. Since 6-holes exist in sufficiently large point sets, only the existence of projective $7$-holes remains open. Similar as in the affine case, Horton sets only have quadratically many projective $k$-holes for $k\le 7$. However, in general the number of $k$-holes can be substantially larger in $R{P}^2$ than in $R^2$ and we construct sets of $n$ points from $R^2\subset R{P}^2$ with $\Omega \left(n^{3-3/5k}\right)$ projective $k$-holes and only $O\left(n^2\right)$ affine $k$-holes for every $k\in \{3,\dots ,$