The compactness theorem for propositional calculus states that a set of propositional sentences has a model (satisfying assignment) if and only if every finite subset of it has a model. I'm looking for uses of the theorem for something outside of logic that are simple enough to present without additional background.
An example is tiling, e.g. tiling of $\mathbb{Z}^2$ using Wang tiles. Given a set of Wang tiles one can introduce variables $X_{i,j}^k$ that means "Tile no. $k$ was placed in $(i,j)$". We now construct an infinite set of sentences saying that the assignment to the variables defines a legal tiling (i.e. for every $(i,j)$ exactly one $X_{i,j}^k$ is true, and we also check edge compatability). Now the compactness theorem can be used to prove that a set of tiles tiles the entire plane if and only if it tiles any finite zone in the plane.
Are there more uses in the same spirit?