application of omitting types theorem in graph theory or some other discrete structure?

Question

i'm looking for some applications of omitting types theorem in discrete structures like graphs... for example something like end extension in PA....

Answer 1

Here's an example application: Suppose $G$ is a graph with the property that for any first-order formula $\varphi(x,y)$ in $2$ free variables, if there are any $a,b\in G$ such that $G\models\varphi(a,b)$, then there are some $a',b'\in G$ such that $G\models\varphi(a',b')$ and there is a path from $a'$ to $b'$. Then there is a connected graph which is elementarily equivalent to $G$.