Erdos-Szekeres type theorems for planar convex sets.

T. Bisztriczky, University of Calgary

November 17, 2015

A family $\mathcal F$ of sets is in convex position if none of its members is contained in the convex hull of the union of the others. The members of $\mathcal F$ are ovals (compact convex sets) in the plane that have a certain property. An Erdős-Szekeres type theorem concerns the existence, for any integer $n\geq3$, of a smallest positive integer $N(n)$ such that if $|\mathcal F|\geq N(n)$ then there are $n$ ovals of $\mathcal F$ in convex position.

In this talk, I survey some known theorems, introduce a new one based upon work with Gábor Fejes Tóth and examine the relation between $N(n)$ and Ramsey numbers of the type $R_k(n_1,n_2)$.