UC San Diego

We define what it means for an equation to be graph-regular, extending the idea of partition- regular equations to a graph setting. An equation is graph -regular if it always has monochromatic solutions under edge-colorings of K/N. We find an infinite family of graph -regular equations, and present two Rado-like conditions which are respectively necessary and sufficient for an equation to be graph-regular. In the process, we prove a Ramsey-like theorem for binary and k-ary trees which may be of independent interest. We also look at a stronger version of Ramsey's theorem from Paris and Harrington, and show a counterexample to the analogous version of van der Waerden's theorem