Is the "or" in Ramsey Theory exclusive? The Ramsey number, $R(n,m)$ is defined to be the order(number of vertices) of the smallest complete graph $G$ such that for any red-blue colouring of $G$ there is a red $K_n$ \emph{or} a blue $K_m$; subgraphs of $G$. My question: Is that "or" exclusive?

No. The word "or" in mathematics is generally not exclusive. If it's meant to be exclusive, it will be specified clearly.

In any case, in a sufficiently large graph, all kinds of subgraphs are likely to occur.

As a trivial example, there exist graphs containing a red $K_n$ and a blue $K_m$ with just $n+m$ vertices (and obviously with any larger number of vertices as well). How can the theorem possible be exclusive, and say that for a graph with sufficently many vertices, it will contain either a red $K_n$ or a blue $K_m$ _but not both_? Such a theorem cannot be true, as we have such counterexamples.