Hindman’s Theorem write-up

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)

1. saf
Posted September 13, 2012 at 9:37 pm | Permalink

Cool! Thanks for posting.

2. Ari Brodsky
Posted October 6, 2012 at 11:16 pm | Permalink

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:
$\{ \{\mathcal U \in U(\mathbb N) : A \in \mathcal U \} : A \subseteq \mathbb N \}$

3. Ari Brodsky
Posted October 7, 2012 at 12:34 am | Permalink

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.

• Ari Brodsky
Posted October 7, 2012 at 12:46 am | Permalink

In #3 I meant semicontinuous.