Hindman’s theorem states that if one colors every natural number either red or blue, then there will be an infinite set X of natural numbers such that all finite sums of distinct elements from X have the same color. The original proof of Hindman’s theorem was a combinatorial mess and the slickest proof is via ultrafilters. In this talk, I will introduce the notion of an ultrafilter on a set, which is simply a division of the subsets of the set into two categories, “small" and “large", satisfying some natural axioms. We will then give the proof of Hindman’s theorem using ultrafilters that are idempotent with respect to a natural addition operation on the set of ultrafilters on the set of natural numbers. Finally, we will introduce an open conjecture of Erdos related to Hindman’s theorem, its reformulation in terms of ultrafilters, and some recent progress made on the problem by myself and my collaborators.
