New Ramsey Classes from Old

Abstract

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be strong amalgamation classes of finite structures, with disjoint finite signatures $\sigma$ and $\tau$. Then $\mathcal{C}_1 \wedge \mathcal{C}_2$ denotes the class of all finite ($\sigma\cup\tau$)-structures whose $\sigma$-reduct is from $\mathcal{C}_1$ and whose $\tau$-reduct is from $\mathcal{C}_2$. We prove that when $\mathcal{C}_1$ and $\mathcal{C}_2$ are Ramsey, then $\mathcal{C}_1 \wedge \mathcal{C}_2$ is also Ramsey. We also discuss variations of this statement, and give several examples of new Ramsey classes derived from those general results.