TITLE: Compactness theorem for propositional calculus - nice uses?
QUESTION [6 upvotes]: 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?

REPLY [5 votes]: Suppose that you have a set $S$ of students and a set $C$ of classes. Each student will be assigned to just one class. Each student $s\in S$ has a finite list $C(s)\subseteq C$ of classes that he or she is willing to take, and each class $c\in C$ has a finite enrolment capacity $n_c$. For $T\subseteq S$, an acceptable course assignment is a function $f:T\to C$ such that 

$f(t)\in C(t)$ for each $t\in T$, and  
$\left|f^{-1}\left[\{c\}\right]\right|\le n_c$ for each $c\in C$.

(In other words, each student gets an acceptable class, and no class is oversubscribed.) 
The compactness theorem now easily implies that if each finite subset of $S$ admits an acceptable course assignment, then so does $S$. Probably the easiest way to model it is as a bipartite graph with parts $S$ and $C$, where each vertex in $S$ is to have degree $1$, each vertex $c\in C$ has degree no more than $n_c$ and for each $s\in S$ the unique edge adjacent to $s$ must terminate at one of the vertices in $C(s)$.