Ernest Schimmerling

Mathematical logic seminar - October 28, 2004

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 "anti-Ramsey theory".

Coloring theorems (more precisely known as negative square-bracket 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 À1 colors in such a way that given an uncountable subset A of ω1 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 ω1.

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.