Abstract: A central question in the theory of ultrahomogeneous relational structures asks, How close of an analogue to the Infinite Ramsey Theorem does it carry? An infinite structure S is ultrahomogeneous if any isomorphism between two finitely generated substructures of can be extended to an automorphism of . We say that has finite big Ramsey degrees if for each finite substructure of , there is a number such that any coloring of the copies of in can be reduced to no more than colors on some substructure of , which is isomorphic to the original .
The two main obstacles to a fuller development of this area have been lack of representations and general Milliken-style theorems. We will present new work proving that the Henson graphs, the-clique free analogues of the Rado graph for , have finite big Ramsey degrees. We devise representations of Henson graphs via strong coding trees and prove Milliken-style theorems for these trees. Central to the proof is the method of forcing, building on Harrington’s proof of the Halpern-Läuchli Theorem.