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)

4 thoughts on “Hindman’s Theorem write-up”

  2. 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:
    \[
    \{ \{\mathcal U \in U(\mathbb N) : A \in \mathcal U \} : A \subseteq \mathbb N \}
    \]

    Like

  3. 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.

    Like

Comments are closed.