 Solution

The compactness theorem for first-order logic says that a set $\Sigma$ of first-order sentences has a model if and only if every finite subset of $\Sigma$ has a model. It applies to arbitrary languages, including expansions by arbitrarily many new constants. Equivalently, if $\Sigma$ entails a sentence, some finite subset already entails it. We will use both formulations with the diagram of a structure.