Series: Penn State Logic Seminar Date: Tuesday, September 16, 2003 Time: 2:30 - 3:45 PM Place: 113 McAllister Building Speaker: Christopher Griffin, Penn State, ARL and Mathematics Title: Coloring Infinite Cardinals: There's not a crayon big enough! Abstract: We will explore colorings on sets constructed from subsets of infinite cardinal numbers. For a fixed natural number n and a set X, [X]^n is the set of subsets of X of cardinality n. A coloring of [X]^n is a function mapping [X]^n to a (possibly infinite) set of colors C_1, C_2, etc. We say that a set H contained in X is homogeneous (with respect to a coloring f) if the coloring f is constant on [H]^n. We will investigate the question: Given a set X, a number n, and a coloring f does there exists a homogeneous set H in X of a given cardinality. Various Ramsey and anti-Ramsey theorems will be proved. In particular, we will discuss the Erdos-Rado theorem and its natural generalizations. We will also introduce compact cardinals and Ramsey cardinals. Time permitting, we will investigate the connections between the existence of homogeneous sets and model theoretic indiscernibles.