Time:  12  1:20 p.m. 
Room: 
Baker Hall 150

Speaker:  Todd Eisworth Department of Mathematics Ohio University 
Title: 
Coloring theorems

Abstract: 
Theodore Motzkin is credited with the observation "Complete disorder is
impossible". In many ways, this maxim captures the essence of Ramsey Theory
 one should search for order hidden among apparent chaos. In this talk,
will be looking at what might be called "antiRamsey theory".
Coloring theorems (more precisely known as negative squarebracket partition relations) assert the existence of functions whose behavior is extremely complicated on every large subset of their domain. As a typical example, there is a theorem of Todorcevic stating that one can color the pairs of countable ordinals with À_{1} colors in such a way that given an uncountable subset A of ω_{1} and given one of our uncountably many colors, there is a pair of ordinals in A that is assigned this color. This coloring exhibits "complicated behavior" (in the sense of taking on all possible values) when restricted to any large subset of ω_{1}.
We will look at some classic results in this area as well as more recent
work, and then speak a bit about open problems. The talk should be
accessible to anyone with a knowledge of basic set theory.
