For the first lecture, see here.
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
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
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.
2. Silver’s theorem.
From the results of the previous lectures, we know that any power can be computed from the cofinality and gimel functions (see the Remark at the end of lecture II.2). What we can say about the numbers varies greatly depending on whether is regular or not. If is regular, then As mentioned on lecture II.2, forcing provides us with a great deal of freedom to manipulate the exponential function at least for regular. In fact, the following holds:
Theorem 1. (Easton). If holds, then for any definable function from the class of infinite cardinals to itself such that:
- whenever and
- for all
there is a class forcing that preserves cofinalities and such that in the extension by it holds that for all regular cardinals here, is the function as computed prior to the forcing extension.
For example, it is consistent that for all regular cardinals (as mentioned last lecture, the same result is consistent for all cardinals, as shown by Foreman and Woodin, although their argument is significantly more elaborate that Easton’s). There is almost no limit to the combinations that the theorem allows: We could have whenever is regular and is an even ordinal, and whenever for some odd ordinal Or, if there is a proper class of weakly inaccessible cardinals (regular cardinals such that ) then we could have the third weakly inaccessible strictly larger than for all regular cardinals etc.
Morally, Easton’s theorem says that there is nothing else to say about the gimel function on regular cardinals, and all that is left to be explored is the behavior of for singular In this section we begin this exploration. However, it is perhaps sobering to point out that there are several weaknesses in Easton’s result.