I have been trying to acclimate myself with finite Ramsey degrees in structural Ramsey theory, my resource being Zucker's TOPOLOGICAL DYNAMICS OF AUTOMORPHISM GROUPS, ULTRAFILTER COMBINATORICS, AND THE GENERIC POINT PROBLEM (https://www.math.cmu.edu/~andrewz/GPP.pdf). Specifically, I have been trying to understand the claim (Proposition 5.8) that if an "excellent" expansion $K^*$ of a Fraisse class $K$ has the Ramsey property, then every element of $K$ has a finite Ramsey degree.
I find it hard, however, to understand the argument. The proof involves a decreasing chain of "thick" sets of embeddings. The existence of a dense set is arguably guaranteed by Lemma 5.9, but I do not see how to guarantee the thick set constructed in each stage is included in the thick set obtained thus far.
What am I missing? I also believe that there is a simpler argument if you assume that each member of $K^*$ is rigid (Zucker does not make this assumption).