Countable homogeneous relational structures have been studied by many people. One area of focus is the Ramsey theory of such structures. After a review of background material, a partition theorem of Laflamme, Sauer, and Vuksanovic for countable homogeneous binary relational structures is discussed with a focus on the size of the set of unavoidable colors.
"Ramsey Theory for Countable Binary Homogeneous Structures." Notre Dame J. Formal Logic 46 (3) 335 - 352, 2005. https://doi.org/10.1305/ndjfl/1125409332