Let $G$ be an infinite graph such that the clique number $\omega(G)$ is finite. By the De Brujin - Erdos theorem, the chromatic number $\chi(G)$ is finite if and only there is a bound on the chromatic number of every finite subgraph. There are constructions of triangle free-graphs with arbitrarily large chromatic numbers, so I cannot hope to deduce $\chi(G) < \infty$ in this general context.
So my question is: what are some sensible conditions on $G$ that ensure that $\chi(G) < \infty$?
For the moment the only sufficient condition I could prove, using the above-mentioned result, is that $\Delta(G) < \infty$ suffices (here $\Delta(G)$ denotes the maximal degree of a vertex of $G$).