Our Results of which some are known
Countable Ordinals and big Ramsey Degrees by Masulovic and Sobot HERE
T(Z choose a) LE 2 to the a.
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 It mentions that Galvin did T(Q choose 2) but this is unpublished, so I am hoping this thesis include that.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.)
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
Big Ramsey Spectra of Countable Chains by Masulovi. HERE