Warning: this proof requires a bit of familiarity with the terminology of propositional logic and graph theory. Problem: Let $ G$ be an infinite graph. Show that $ G$ is $ n$-colorable if and only if every finite subgraph $ G_0 \subset G$ is $ n$-colorable. Solution: One of the many equivalent versions of the Compactness Theorem for the propositional calculus states that if $ \Sigma \subset \textup{Prop}(A)$, where $ A$ is a set of propositional atoms, then $ \Sigma$ is satisfiable if and only i...