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.