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*)

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Cool! Thanks for posting.

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On page 3, the description of the “basis of open sets” for the topology on 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 \}

\]

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The proof of Theorem 3.1 relies on several unstated facts that should be explicitly stated (if not proven) in the exposition:

1) is compact.

2) is Hausdorff. [Note that is not enough, because we need all compact subsets to be closed. In fact there is a compact, topological semigroup having no idempotents, namely with the minimal topology, also known as the finite complement topology.]

3) The operation is continuous.

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In #3 I meant semicontinuous.

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