Tae Young Lee, University of Texas, Austin

Abstract

Imagine five points in R^2, where no three of them are colinear. You can always find a convex quadrilateral among them. How many points do you need for a convex pentagon? What about a convex k-gon? Is it even possible? Erdős and Szekeres proved that this is indeed possible whenever you have at least ES(k) points in general position, where ES(k) is some number not exceeding ((2k-4) choose (k-2))+1. They conjectured that ES(k)=2^{k-2}+1, and later proved that this is a lower bound. I will present their proofs about these facts and a sketch of the proof of the best known upper bound by Andrew Suk. If time permits, I will also briefly discuss some variants and generalizations of this problem.

Presented via Zoom: https://rutgers.zoom.us/j/98441409199

Password: 715004

See: https://sites.math.rutgers.edu/~qcd2/GCS.html