Ramsey Degrees

Our results (some are already known)

Our Results of which some are known

nathan

Survey with no proofs

When Ramsey Theory Fails …

Z

Countable Ordinals and big Ramsey Degrees by Masulovic and Sobot HERE

T(Z choose a) LE 2 to the a.

Q

Some Partition Theorems and Ultrafilters on Omega by Denis Devlin HERE

This is a PhD thesis that has many results, including material on T(Q choose a). It mentions that Galvin did T(Q choose 2) but this is unpublished, so I am hoping this thesis include that. See next two entries.

A proof of a partition theorem for Q to the n by Vojan Vuksanovic HERE

This claims to be an easier proof of the main theorems in Devlin's thesis; however, I do not know what he is proving.

A partition theorem by Halpern and Lauchli. HERE

This papers motivation is stuff we don't care about, but as a lemma they prove a Ramsey Theorem on tress that is used by Devlin and I think others, on our kind of problems. This proof is difficult. There are easier proofs in the following sources:

Introduction to Ramsey Spaces by Todorcevic HERE

Ramsey Theory for Product Spaces by Dodosa and Kanellopoulos. HERE

Some Appliations of Forcing by Todorceic and Farah. (This is a book that is not online.)

Ordinals

Countable Ordinals and big Ramsey Degrees by Masulovic and Sobot HERE

Let ALPHA be a countable ordinal. They show that

(forall n)[T(ALPHA choose n) is finite IFF ALPHA LL omega omega

Scattered Linear Orderings

Big Ramsey Spectra of Countable Chains by Masulovi. HERE

HERE