Suppose $G=(V,E)$ is a simple but not necessarily finite graph, and let $E',E''$ be a disjoint partitioning of $E$ into two (not necessarily finite) subsets, so that $G$ is the edge disjoint union of the graphs $G'=(V,E')$ and $G''=(V,E'')$.
My (admittedly vague) question is: when can the maximum clique size of $G$ be related to the maximum clique sizes of $G'$ and $G''$? In the case that $G$ is infinite, is it even clear that maximum clique size of $G$ and $G'$ both finite implies that the maximum clique size of $G$ is finite?
I'm guessing that in complete generality probably little can be said, because a clique of $G$ need not a priori respect the clique structures of $G'$ and $G''$. But in my setting, I have a particular pair of graphs $G', G''$ that I know a little bit about- both are infinite and locally infinite, both have finite maximum cliques (and I know these sizes explicitly), both have finite chromatic number, they're bi-lipschitz equivalent to each other, etc. So I'd also be happy with results that assume extra structure (but not finiteness) on $G,G',G''$.
Thanks for reading; any ideas would be greatly appreciated.