Our Results of which some are known

*Countable Ordinals and big Ramsey Degrees*
by Masulovic and Sobot
HERE

They prove . We have easier proof in our ordinal paper.

*Some Partition Theorems and Ultrafilters on Omega*
PhD by Denis Devlin
HERE

This contains much about . It cites Galvin (unpublished) but this is unpublished. I can't tell if it has the exact formula for .

*A proof of a partition theorem for *
by Vojan Vuksanovic HERE

Easier proof of the main theorems in Devlin's thesis. It does have the exact formula for , though I do not know if Devlin also did that.

*A partition theorem*
by Halpern and Lauchli.
HERE

This paper proves a Ramsey Theorem on trees that is NOT about Ramsey Degrees but is used in many later papers. The 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 be a countable ordinal. They show: is IFF .

*Big Ramsey Spectra of Countable Chains* by Masulovi. HERE

Let be a Scattered Linear Order.

*An Order Type Decomposition Theorem* by Richard Laver HERE
.

*Big Ramsey Spectra of countable chains* by Dragan Masulovic HERE

If has finite Hasudroff rank and bounded finite sums then .

If has infinite Hasudroff rank then .

is the Rado -ary Hypergraph.

The notion of finite Ramsey Degree is much more general here, looking at coloring finite substructures of the graph.

*Edge Partitions of the Rado Graph*
by Pouzet and Sauer.
HERE

They prove that for all finite colorings of the EDGES of the Rado graph there is an isomorphic copy of which only uses 2 colors. We can cal this . See next paper for .

*Canonical Partitions of Universal Structures*
by Laflamme, Sauer, Vuksanovic
HERE
They show how to computer exactly though do not give any numbers
explicitly.

*Counting Canonical Parttions in the Random Graph*
by Jean Larson
HERE
They give an algorithm to computer .

*Strong Embeddings of Graphs into Colored Graphs*
By Erdos, Hajnal, Posa.
HERE

*Coloring Subgraphs of the Rado Graph*
by Sauer
HERE
They prove that there exists a finite coloring of the EDGES of the Rado graph
such that any isomorphic copy of uses colors.

*Big Ramsey Degrees of 3-Uniform Hypergraphs are Finite*
by Balko, Chodounsky, Hubicka,Konecny, Vena
HERE

They show that has finite big Ramsey degree.

*The Ramsey Theory of Henson Graphs*
by Dobrinen
Here

The Henson graph is -clique free analogue of the Rado graph. This paper shows that has finite big Ramsey Degree.

*Big Ramsey Degree and Topological Dynamcs*
By Andy Zucker.
HERE

*Finite Big Ramsey Degree in Universal Structures*
By Masulovic
HERE

*Ramsey degrees of finite ultrametric spaces, ultrametric Urysohn spaces,
and dynamics of their isometry groups*
By Lionel Nguyen Van The
HERE

*Big Ramsey Degrees using Paramter Spaces*
By Jan Hubicka
HERE