Finite Intersection Propery – Mike Pawliuk

Stevo’s Forcing Class Fall 2012 – Class 5

(This is the fifth lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the fourth lecture here. Quotes by Stevo are in dark blue; some are deep, some are funny, some are paraphrased so use your judgement. As always I appreciate any type of feedback, including reporting typos, in the comments below.)

Continue reading Stevo’s Forcing Class Fall 2012 – Class 5

Stevo’s Forcing Class Fall 2012 – Class 4

(This is the fourth lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the third lecture here. Quotes by Stevo are in dark blue; some are deep, some are funny, some are paraphrased so use your judgement. As always I appreciate any type of feedback, including reporting typos, in the comments below.)

Continue reading Stevo’s Forcing Class Fall 2012 – Class 4

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)

MA and its effect on Tree Partitions

(This is the presentation I gave for Stevo Todorcevic’s course Combinatorial Set Theory on Feb 28, 2012. The material comes from Stevo’s 1983 paper “Partition Relations for Partially Ordered Sets”.)

In partition relations for ordinals, it has been established that:

Theorem (Erdos-Rado). $\omega_1 \rightarrow (\omega_1, \omega+1)^2$

Later it was shown that this is the best you can do, as the strengthenings are consistent:

Theorem(Hajnal). Under CH, $\omega_1 \not\rightarrow (\omega_1, \omega+2)^2$
Theorem (Todorcevic). Under PFA, for any countable ordinal $\alpha$ , $\omega_1 \rightarrow (\omega_1, \alpha)^2$

Moving on, we can ask the same questions about non-special trees, which in some way are the tree analogue of “uncountable” or “large”.

Theorem (Todorcevic). Nonspecial Tree $\rightarrow$ (Nonspecial Tree, $\omega+1)^2$

This is the analogue or the Erdos-Rado theorem.

Recall that a tree $T$ is nonspecial if $T \rightarrow (\omega)^1_\omega$ , which means that any countable partition $T$ contains an infinite set. (This is a generalization of uncountable, because for countable sets you can always put one element per colour.)

We will show the following:

Theorem (Todorcevic). Under MA, for $T$ a tree with no uncountable chains and $\vert T \vert = 2^{\aleph_0}$ we have $T \not\rightarrow (T, \omega+2)^2$ .

Continue reading MA and its effect on Tree Partitions

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)

My first entry! “Helly’s Theorem”

I love compactness. I really do. It turns infinite things into (almost) finite things. I could gush about how great it is, but instead let me tell you about one problem where compact sets act as the delimiter.

Here is one way to characterize compactness:

A space X is compact if and only if any family of closed sets with the Finite Intersection Property (FIP) has a common point.

[Remember that a collection has the FIP if every finite subcollection has a common point (i.e. has non-empty intersection).]

This has a pretty clear connection to filters, as filters are collections of sets with the FIP (and the intersection is in the filter!) and closed under supersets. One example of a filter is the collection of all subsets of the real line that contain a closed interval around 0.

A closely related notion is that of being 2-linked. A collection A is 2-linked if any two sets in A have non-empty intersection. For example the collection of real intervals $\{(n,n+2): n \in \mathbb{Z}\}$ is 2-linked. Another example is the set of sides of a ~~polygon~~ triangle. (Why not a square?)

Then of course we can talk about being 3-linked which means that any 3 sets have non-empty intersection (we will now say that this is called ‘meeting’). Obviously, $\{(n,n+2): n \in \mathbb{Z}\}$ is 2-linked, but not 3-linked. (edit: Yeah, so not only is this not obvious, but it is not true! I address this here.)

Then we could go on to define n-linked for an arbitrary natural number n.

Question 1: How is the FIP related to being n-linked?
Question 2: Can you find, for each n, an example of a collection that is n-linked but not n+1 linked?
Question 3: How is n-linkedness related to the dimension of the real line?

I’ll get to these later. But you should think about them. 1 and 2 are not hard. 3 takes some thought, but just try to come up with a conjecture.

Continue reading My first entry! “Helly’s Theorem”