Let me begin with a couple of updates.
In the last Corollary of the Appendix to lecture I.5, I indicate that in we have that
whenever is not for some infinite limit ordinal In fact,
This result is best possible in terms of positive results. In Theorem 11 of the paper by John Hickman listed at the end, it is shown that for any such it is consistent with that there is an with for which
I also want to give an update on the topics discussed in lecture III.3.
and Hajnal asked whether it is possible to have infinite cardinals such that
Galvin and Prikry showed (see Corollaries 16 and 18 of lecture III.3) that any such must be larger than and that
Following a nice suggestion of Grigor Sargsyan, we use arguments as in Theorem 9 from lecture III.5 to show that this partition relation cannot hold.
The key is the following:
Lemma 1 If there are infinite cardinals such that then for every sufficiently large there is an elementary embedding such that and
Here is a brief sketch:
Proof: By Corollary 20 from lecture III.3, the given relation is equivalent to Consider a -Skolem function so that any closed under is both closed under -sequences and an elementary substructure of
Use to define a coloring by setting whenever and otherwise. By assumption, there is with Note that if is the closure of under then But we can assure that and the result follows by taking as the transitive collapse of
One concludes the proof by noting that it is impossible to have such embeddings. For this, it suffices that and that admits a fixed point past its critical point. One then obtains a contradiction just as in Kunen’s proof that there are no embeddings see Corollary 9 in lecture III.3.
Similarly, Matthew Foreman has shown that there are no embeddings with closed under -sequences. The reason is that any such embedding must admit a fixed point past its critical point, as can be argued from the existence of scales. See the paper by Vickers and Welch listed at the end for a proof of this result.
On the other hand, it is still open whether one can have embeddings such that computes cofinality correctly.
1. The Baumgartner-Hajnal theorem
In Theorem 2 of lecture III.5 we showed the -Rado result that
whenever is regular. It is natural to wonder whether stronger results are possible. We restrict ourselves here to the case Due to time constraints, we state quite a few results without proof.