(continued from August 30th)

Hindman's Theorem asserts that whenever the natural numbers are colored with finitely many numbers there is an infinite set $A$ of natural numbers such that all finite nonempty sums of distinct elements of $A$ have the same color. I will discuss joint work with Barbara Csima, Damir Dzhafarov, Denis Hirschfeldt, Reed Solomon, and Linda Westrick in which we consider the effective analysis of a very special case of Hindman's Theorem where only sums of length exactly two are considered. This is simultaneously a special case of Ramsey's Theorem for pairs. Our main result is that there is a computable 2-coloring of the natural numbers such that there is no infinite Sigma02 set $A$ such that all sums $a + b$ with $a, b$ distinct elements of $A$ have the same color. The proof is based on the paper "Probabilistic constructions of computable objects and a computable version of Lovász Local Lemma" by Andrei Rumyantsev and Alexander Shen. I will give an overview of this paper and will not assume that those in attendance are already familiar with the probabilistic method.