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 \(\mathbf S\) is ultrahomogeneous if any isomorphism between two finitely generated substructures of \(\mathbf S\) can be extended to an automorphism of \(\mathbf S\). We say that \(\mathbf S\) has finite big Ramsey degrees if for each finite substructure \(A\) of \(\mathbf S\), there is a number \(n(A)\) such that any coloring of the copies of \(A\) in \(\mathbf S\) can be reduced to no more than \(n(A)\) colors on some substructure \(\mathbf S'\) of \(\mathbf S\), which is isomorphic to the original \(\mathbf S\).

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 \(k\)-clique free analogues of the Rado graph for \(k\ge 3\), have finite big Ramsey degrees. We devise representations of Henson graphs via strong coding trees and prove Millike-style theorems for these trees. Central to the proof is the method of forcing, building on Harrington's proof of the Halpern-Läuchli Theorem.

There is a video recording of this talk on YouTube.

Slides are available, too.