Title: The Ramsey theory of the universal homogeneous triangle-free graph,
Abstract: 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 Erdős-Hajnal-Posá 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 (Komjáth-Rödl) 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.