Andrew Hoon Suk
Geometric Ramsey Theory
Abstract: By Ramsey's theorem, any system of n segments in the plane has roughly logn members that are either pairwise disjoint or pairwise intersecting. Analogously, any set of n points p(1),..., p(n) in the plane has a subset of roughly loglogn elements with the property that the orientation of p(i)p(j)p(k) is the same for all triples from this subset with i less than j less than k. (The elements of such a subset form the vertex set of a convex polygon.) However, in both cases we know that there exist much larger "homogeneous" subsystems satisfying the above conditions. What is behind this favorable behavior? In the second problem, it is known that the edge set in the underlying hypergraph has a transitive-like property. In both problems, the underlying graphs and hypergraphs are "semi-algebraic", that is, they can be defined by a small number of polynomial equations and inequalities in terms of the coordinates of the segments and points. In this talk, I will discuss several joint results with Conlon, Fox, Pach, and Sudakov, establishing new Ramsey-type bounds for such hypergraphs defined geometrically.
Wednesday January 16, 2013 at 2:00 PM in SEO 636