  Ramsey theory and forcing have a symbiotic relationship. At the RIMS
Symposium on Infinite Combinatorics and Forcing Theory in 2016, the author gave
three tutorials on Ramsey theory in forcing. The first two tutorials
concentrated on forcings which contain dense subsets forming topological Ramsey
spaces. These forcings motivated the development of new Ramsey theory, which
then was applied to the generic ultrafilters to obtain the precise structure
Rudin-Keisler and Tukey orders below such ultrafilters. The content of the
first two tutorials has appeared in an expository article submitted to the
SEALS 2016 Proceedings. The third tutorial concentrated on uses of forcing to
prove Ramsey theorems for trees which are applied to determine big Ramsey
degrees of homogeneous relational structures. This is the focus of this paper.
