Natasha Dobrinen (University of Denver) Henson's universal trianglefree graphs have finite big Ramsey degrees Abstract: A trianglefree graph on countably many vertices is universal trianglefree if every countable trianglefree graph embeds into it. Universal trianglefree graphs were constructed by Henson in 1971, which we will denote as $\mathcal{H}_3$. Being an analogue of the random graph, its Ramsey properties are of interest. Henson proved that for any partition of the vertices in $\mathcal{H}_3$ into two colors, there is either a copy of $\mathcal{H}_3$ in one color (furthermore, only leaving out finitely many vertices in the first color), or else the other color contains all finite trianglefree graphs. In 1986, Komj\'{a}th and R\"{o}dl proved that the vertices in $\mathcal{H}_3$ have the Ramsey property: For any partition of the vertices into two colors, one of the colors contains a copy of $\mathcal{H}_3$. In 1998, Sauer showed that there is a partition of the edges in $\mathcal{H}_3$ into two colors such that every subcopy of $\mathcal{H}_3$ has edges with both colors. He also showed that for any coloring of the edges into finitely many colors, there is a subcopy of $\mathcal{H}_3$ in which all edges have at most two colors. Thus, we say that the big Ramsey degree for edges in $\mathcal{H}_3$ is two. It remained open whether all finite trianglefree graphs have finite big Ramsey degrees; that is, whether for each finite trianglefree graph G there is an integer n such that for any finitary coloring of all copies of G, there is a subcopy of $\mathcal{H}_3$ in which all copies of G take on no more than n colors. We prove that indeed this is the case. 
