It came to my attention that Leo Goldmakher had written up notes for a lecture I gave in August 2011 on the proof of Hindman’s Theorem via ultrafilters. The notes are quite nice so I thought I would share them.
Here is a link to the notes (pdf) and here is Leo’s website.
The lecture I gave follows the papers:
- “An Algebraic Proof of van der Waerden’s Theorem” by Vitaly Bergelson, Hillel Furstenburg, Neil Hindman and Yitzhak Katznelson. (L’enseignement Mathematique, t. 35, 1989, p. 209-215)
- “Ultrafilters: Some Old and some New Results” (pdf) by W.W. Comfort. (Bulletin of the AMS, Volume 83, Number 4, July 1977)
4 Comments
Cool! Thanks for posting.
On page 3, the description of the “basis of open sets” for the topology on $U(\mathbb N)$ is not quite right. Describing a collection of open sets requires another set of braces:
\[
\{ \{\mathcal U \in U(\mathbb N) : A \in \mathcal U \} : A \subseteq \mathbb N \}
\]
The proof of Theorem 3.1 relies on several unstated facts that should be explicitly stated (if not proven) in the exposition:
1) $U(\mathbb N)$ is compact.
2) $U(\mathbb N)$ is Hausdorff. [Note that $T_1$ is not enough, because we need all compact subsets to be closed. In fact there is a compact, $T_1$ topological semigroup having no idempotents, namely $(\mathbb N, +)$ with the minimal $T_1$ topology, also known as the finite complement topology.]
3) The operation $\oplus$ is continuous.
In #3 I meant semicontinuous.