Tag: Ramsey Theory

Van der Waerden’s Theorem is false for $latex \omega_1 $

One of my main research problems involves something I think is related to arithmetic progressions in \mathbb{Z} . After learning a nice proof of Van der Waerden’s Theorem, and sharing it with colleagues, Daniel Soukup asked a nice question about Van der Waerden’s Theorem on \omega_1 . We answered it, and the example was sufficiently nice that I would like to share it.

Continue reading Van der Waerden’s Theorem is false for \omega_1 

Van der Waerden’s Theorem is false for $latex omega_1 $

One of my main research problems involves something I think is related to arithmetic progressions in mathbb{Z} . After learning a nice proof of Van der Waerden’s Theorem, and sharing it with colleagues, Daniel Soukup asked a nice question about Van der Waerden’s Theorem on omega_1 . We answered it, and the example was sufficiently nice that I would like to share it.

Continue reading Van der Waerden’s Theorem is false for omega_1 

Another Combinatorial Result

Here is Chris Eagle’s presentation from Stevo Todorcevic’s class “Combinatorial Set Theory”.

From the abstract:

We prove that MA + \mathfrak{c} = \aleph_2 implies \mathfrak{c} \not\rightarrow (\mathfrak{c}, \omega+2)^2 . The exposition is based on hand-written notes provided by S. Todorcevic. The result itself is due to R. Laver.

This is the analogous result to “MA implies (NonSpecial Tree) \not\rightarrow (NonSpecial Tree, \omega+2)^2 “, which I explained here.

The Delta-System Lemma

colorado_river_delta

Now that you know the basics of countable elementary submodels (CESM), you might think that you are in the clear. “Mike”, you say arrogantly, “I know all the most basic properties of CESMs, without proof I remind you, what else could I possibly want?”. I gently and patiently remind you that CESMs are worthless unless you know how to apply them properly.

So let’s do that.

Here are two theorems whose proofs you might already know, but that can be proved using elementary submodels. I will show you a proof of the \Delta -system lemma (a fundamental lemma in infinitary combinatorics) and a topological theorem of Arhangel’skii. Both of these proofs are taken from Just & Weese’s book “Discovering Modern Set Theory 2”, chapter 24.

Continue reading The Delta-System Lemma

Helly’s Theorem (2/2)

Last week we looked at the concepts of a collection of sets being n-linked or having the finite intersection property. The key theorem was Helly’s theorem which says:

Helly’s Theorem: If a (countable) family of closed convex sets (at least one of which is bounded) in the plane are 3-linked, then they have a point in common, as they have the FIP.

Now I will look at some of the generalizations that Alexander Soifer, author of “The Mathematical Coloring Book”, makes in Chapter 28 of that book. More than pure generalizations they are the combination of Ramsey theory and Helly’s Theorem

Continue reading Helly’s Theorem (2/2)