The Ramsey theory of the universal homogeneous triangle-free graph

The universal homogeneous triangle-free graph, constructed by Henson and denoted ${\mathcal{H}}_3$, is the triangle-free analogue of the Rado graph. While the Ramsey theory of the Rado graph has been completely established, beginning with Erd\H{o}s-Hajnal-Pos'{a} and culminating in work of Sauer and Laflamme-Sauer-Vuksanovic, the Ramsey theory of ${\mathcal{H}}_3$ had only progressed to bounds for vertex colorings (Komj'{a}th-R"{o}dl) and edge colorings (Sauer). This was due to a lack of broadscale techniques. We solve this problem in general: For each finite triangle-free graph $G$, there is a finite number $T(G)$ such that for any coloring of all copies of $G$ in ${\mathcal{H}}_3$ into finitely many colors, there is a subgraph of ${\mathcal{H}}_3$ which is again universal homogeneous triangle-free in which the coloring takes no more than $T(G)$ colors. This is the first such result for a homogeneous structure omitting copies of some non-trivial finite structure. The proof entails developments of new broadscale techniques, including a flexible method for constructing trees which code ${\mathcal{H}}_3$ and the development of their Ramsey theory.