Assumption: For any graph $H$ there is a graph $G$ where any 2-coloring of $G$ either has $H$ colored 1 or $H$ colored 2.
Want to show: For any graphs $H_1,H_2$ there is a graph $G$ where any 2-coloring of $G$ either has $H_1$ colored 1 or $H_2$ colored 2.
I was thinking that we have Ramsey Graphs $G_1$ and $G_2$ for $H_1$ and $H_2$, and if we take $G_1\cup G_2$ the only case we need to consider is when $H_1$ is colored 2 and $H_2$ is colored 1. Then somehow I want to force there to be $H_1$ colored 1 or $H_2$ colored 2 from this assumption.