Special Colloquium
Andrew Hoon Suk
MIT
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