While not strictly set theory, but rather a result connecting Ramsey theory and topological dynamics, two areas that are rather close to set theory, you might find the following example interesting:

In http://arxiv.org/pdf/math/0305241.pdf Kechris, Pestov and Todorcevic showed that a Fraisse class of finite structure is Ramsey iff the automorphism group of the Fraisse limit is extremely amenable.

Let me explain: The Fraisse limits that we are talking about here are countable structures and the automorphism groups of these structures carry a natural topology, the topology of pointwise convergence, which in this situation is separable and metric. Given a topological group $G$, a continuous action of $G$ on a compact space $X$
is minimal if every orbit is dense in $X$. We call a compact space $X$ together with a continuous $G$-action a $G$-flow. Using Zorn's lemma, we see that every $G$-flow contains a minimal $G$-flow. It turns out that there is a universal minimal $G$-flow, i.e., one that maps onto every other minimal $G$-flow by a $G$-equivariant map (a map that respects the group action). Now the construction of the universal minimal flow can be phrased in various ways, but concepts like ultrafilters and other applications of Zorn's lemma show up here.
So this is, in some sense, set theory.

Now a topological group is extremely amenable if its universal minimal flow has just one point, or equivalently, if every $G$-flow has a fixed point.

On the other hand, a class of finite structures is Ramsey if for all $A$ and $B$ in the class there is $C$ in the class such that for every coloring of all copies of $A$ inside $C$ with two colors, there is a copy $B'$ of $B$ in $C$ such that all copies of $A$ in $B'$ have the same color. The usual Ramsey theorem shows that the class of finite linear orders is a Ramsey class and the Nesetril-Rödl theorem is precisely the statement that the class of finite ordered graphs is a Ramsey class.

As corollaries from the Kechris, Pestov, Todorcevic theorem we get that the automorphism groups of the countable dense linear order without endpoint $(\mathbb Q,\le)$ and of the ordered random graph are extremely amenable.