In 1935, Erdős and Szekeres posed (and answered) the following question. For each n>2, is it possible to put enough points in general position in the plane so that it is guaranteed that some n-subset of these points forms a convex n-gon? This question is indeed true, and the exact number of points needed was conjectured by Erdős and Szekeres in 1935. Later, natural generalizations of this question were formed in higher dimensions. We will begin with the base cases of this problem in two and three dimensions. Then, we will discuss a purely graph theoretic Ramsey argument that immediately proves the existence of such point sets. Finally, we will explore geometric cup-cap arguments that have been used to provide tighter bounds on the size of these point sets.
Zoom Link: https://washington.zoom.us/j/92849568892