580 -Partition calculus (6)

 

1. The -Rado theorem

 

Large homogeneous sets (of size or larger) can be ensured, at the cost of starting with a larger domain. The following famous result was originally shown by and Rado using tree arguments (with lowered to in the conclusion). We give instead an elementary substructures argument due to Baumgartner, Hajnal and which proves the stronger version. For a cardinal let

Theorem 1 (-Rado) Let be a regular cardinal and let Then

for all

 

305 -Homework set 8

This set is due May 1 at the beginning of lecture. Details of the homework policy can be found on the syllabus and here.

1. Solve exercises 11, 14, and 24 from Chapter 14 of the textbook.

2. Solve exercises 1, 9, and 17 from Chapter 15 of the textbook.

305 -9. Groups

We want to define the notion of group that will be fundamental to determine which polynomials are solvable by radicals. This notion is very important and appears in every area of mathematics.

We motivate the definition through the example that most concerns us: automorphisms of fields. They are particular class of isomorphisms, so we begin with them.

Some of the arguments below have been discussed in previous lectures.

580 -Partition calculus (5)

1. Larger cardinalities

We have seen that (Ramsey) and (-Rado) for any On the other hand, we also have that () and (-Kakutani) for any infinite

Positive results can be obtained for larger cardinals than if we relax the requirements in some of the colors. A different extension, the -Rado theorem, will be discussed later.

Theorem 1 (-Dushnik-Miller) For all infinite cardinals

This was originally shown by Dushnik and Miller in 1941 for regular, with providing the singular case. For regular one can in fact show something stronger:

Theorem 2 (-Rado) Suppose is regular and uncountable. Then

which means: If then either there is a stationary that is -homogeneous for , or else there is a closed subset of of order type that is -homogeneous for .

(Above, top stands for “topological.”)

305 -8. Irreducibility

In this lecture I want to present a couple of short results that are nevertheless very useful in practice when trying to show that a given polynomial in is irreducible. Of course, we may assume that the polynomial actually has integer coefficients. In this case, it turns out that analyzing whether the polynomial factors over suffices.

580 -Cardinal Arithmetic

Here is a pdf file with the contents of the lectures on cardinal arithmetic. As with the previous chapter, it follows closely the style of the notes. There are fewer typos than in the posts, and once again I made a minuscule tidying up. Please let me know of comments, corrections, and suggestions.

580 -Some choiceless results

I reformatted the posts for the lectures on choiceless results. Here they are as a pdf file. It follows closely the style of the notes. I fixed a couple of typos, and made a minuscule tidying up. Please let me know of comments, corrections, and suggestions.

Update (Dec. 3, 2009). The pdf file still contains an error that was pointed out to me recently. In the third lecture on choiceless results, the correct argument is given.

305 -Extension fields revisited (3)

1. Isomorphisms

We return here to the quotient ring construction. Recall that if is a commutative ring with identity and is an ideal of then is also a commutative ring with identity. Here, where for the equivalence relation defined by iff

Since is an equivalence relation, we have that if and if In particular, any two classes are either the same or else they are disjoint.

In case for some field then is principal, so for some i.e., given any polynomial iff and, more generally, (or, equivalently, or, equivalently, ) iff

In this case, contains zero divisors if is nonconstant but not irreducible.

If is 0,

If is constant but nonzero, then

Finally, we want to examine what happens when is irreducible. From now on suppose that this is the case.

305 -Extension fields revisited (2)

Most of our work from now on depends on the following simple, but very useful observation:

Theorem 1 Let be a field extension. Then with its usual addition is a vector space over where multiplication of elements of by elements of is just the usual product of .

580 -Partition calculus (4)

 

1. Colorings of pairs. I

 

There are several possible ways in which one can try to generalize Ramsey’s theorem to larger cardinalities. We will discuss some of these generalizations in upcoming lectures. For now, let’s highlight some obstacles.

Theorem 1 (-Kakutani) In fact,

 

Proof: Let Let be given by

Then, if are distinct, it is impossible that

Theorem 2 (Sierpi\’nski) In fact,

 

Proof: With as above, let be given as follows: Let be a well-order of in order type Let be the lexicographic ordering on Set

Lemma 3 There is no -increasing or decreasing -sequence of elements of

 

Proof: Let be a counterexample. Let be least such that has size and let be such that if then To simplify notation, we will identify and For let be such that but By regularity of there is such that for many

But if and then iff so It follows that has size contradicting the minimality of

The lemma implies the result: If has size and is -homogeneous, then contradicts Lemma 3.

Now I want to present some significant strengthenings of the results above. The results from last lecture exploit the fact that a great deal of coding can be carried out with infinitely many coordinates. Perhaps surprisingly, strong anti-Ramsey results are possible, even if we restrict ourselves to colorings of pairs.