For the first-order language with vocabulary $(E)$ (the binary relation $E$ which holds if two vertices have an edge) together with a set $G$ of vertices, I've been told that the property "a symmetric graph is connected" cannot be axiomatized by any set of first-order sentences.
I think the proof involved taking two constants, let's say $x,y \in G $ and building an infinite set of sentences $T$ that states "there is no path of length $n$ between $x$ and $y$" for each $n$ (where a path is, for some vertices $(v_1,...,v_n) \in G$, $[E(x, v_1) \wedge ... \wedge E(v_n, y)]$). By compactness, I see that $T$ is satisfiable. However, I don't see how $T$ shows the impossibility of constructing another set of sentences $T'$ which is satisfied by connected graphs.
I do understand the use of compactness to show that a theory whose models have arbitrarily large domains also has a model with an infinite domain, but I don't understand its use here. I don't have any experience with ultraproducts, so answers using that concept may be lost on me.