The Infinite Ramsey Theorem states that given $n, r \ge 1$ and a coloring of all $n$-sized subsets of $\mathbb{N}$ into $r$ colors, there is an infinite subset of $\mathbb{N}$ in which all $n$-sized subsets have the same color. Extensions of Ramsey's Theorem to homogeneous structures have been studied for several decades, in particular infinite graphs. In this setting, one colors all copies of a finite graph within a given infinite graph, the goal being to find an infinite subgraph isomorphic to the original one in which as few colors as possible are used. This question gained new momentum when it was brought into focus by Kechris, Pestov, and Todorcevic (KPT) in their groundbreaking work on the correspondence between the Ramsey property and extreme amenability. Answering a question of KPT, Zucker found a correspondence between Ramsey degrees for infinite structures and completion flows, providing additional motivation for the search for infinite structures with good Ramsey properties.

We present {\em strong coding trees} and their Ramsey theory. These were developed by the speaker in order to find upper bounds for the Ramsey degrees of the random $k$-clique-free graphs. However, it turns out that strong coding trees are also useful for coding copies of the Rado graph and other homogeneous structures. We will give an overview of how these methods are used

to find upper bounds for the Ramsey degrees of the random $k$-clique-free graphs, and some infinite dimensional Ramsey theory of the Rado graph, answering a question implicit in KPT.