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

Description of video

Date: 4/22/22
Speaker :Scheucher Manfred

In this talk 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 M\"oller in 1994. The notion of convex position in RP2 agrees with the definition of convex sets introduced by Steinitz in 1913. An affine k-hole in a finite set SR2 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 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 k7. However, in general the number of k-holes can be substantially larger in RP2 than in R2 and we construct sets of n points from R2RP2 with Ω(n33/5k) projective k-holes and only O(n2) affine k-holes for every