In this talk we consider point sets in the real projective plane RP2$\mathbb{R}{P}^{2}$ 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 RP2$\mathbb{R}{P}^{2}$, which was initiated by Harborth and M\"oller in 1994. The notion of convex position in RP2$\mathbb{R}{P}^{2}$ agrees with the definition of convex sets introduced by Steinitz in 1913. An affine k$k$-hole in a finite set S⊆R2$S\subseteq {\mathbb{R}}^{2}$ is a set of k$k$ points from S$S$ in convex position with no point of S$S$ in the interior of their convex hull. After introducing a new notion of k$k$-holes for points sets from RP2$\mathbb{R}{P}^{2}$, called projective k$k$-holes, we find arbitrarily large finite sets of points from RP2$\mathbb{R}{P}^{2}$ 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$7$-holes remains open. Similar as in the affine case, Horton sets only have quadratically many projective k$k$-holes for k≤7$k\le 7$. However, in general the number of k$k$-holes can be substantially larger in RP2$\mathbb{R}{P}^{2}$ than in R2${\mathbb{R}}^{2}$ and we construct sets of n$n$ points from R2⊂RP2${\mathbb{R}}^{2}\subset \mathbb{R}{P}^{2}$ with Ω(n3−3/5k)$\mathrm{\Omega}({n}^{3-3/5k})$ projective k$k$-holes and only O(n2)$O({n}^{2})$ affine k$k$-holes for every k∈{3,…,