The Ramsey theory of Henson graphs

For $k\ge 3$, the Henson graph $\mathcal{H}_k$ is the analogue of the Rado graph in which $k$-cliques are forbidden. Building on the author's result for $\mathcal{H}_3$, we prove that for each $k\ge 4$, $\mathcal{H}_k$ has finite big Ramsey degrees: To each finite $k$-clique-free graph $G$, there corresponds an integer $T(G,\mathcal{H}_k)$ such that for any coloring of the copies of $G$ in $\mathcal{H}_k$ into finitely many colors, there is a subgraph of $\mathcal{H}_k$, again isomorphic to $\mathcal{H}_k$, in which the coloring takes no more than $T(G, \mathcal{H}_k)$ colors. Prior to this article, the Ramsey theory of $\mathcal{H}_k$ for $k\ge 4$ had only been resolved for vertex colorings by El-Zahar and Sauer in 1989. We develop a unified framework for coding copies of $\mathcal{H}_k$ into a new class of trees, called strong $\mathcal{H}_k$-coding trees, and prove Ramsey theorems for these trees, forming a family of Halpern-\Lauchli\ and Milliken-style theorems which are applied to deduce finite big Ramsey degrees. The approach here streamlines the one in \cite{DobrinenH_317} for $\mathcal{H}_3$ and provides a general methodology opening further study of big Ramsey degrees for homogeneous structures with forbidden configurations. The results have bearing on topological dynamics via work of Kechris, Pestov, and Todorcevic and recent work of Zucker.