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 It mentions that Galvin did T(Q choose 2) but this is unpublished, so I am hoping this thesis include that.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