I think one could make the case that idempotent ultrafilters are so closely related to Ramsey theory that anything which uses them does relate to Ramsey theory in some way (almost by definition). A less obvious example though would be Gowers's result that $c_0$ is oscillation stable --- that every bounded Lipschitz function on the sphere of $c_0$ is $\epsilon$-constant when restricted to an infinite dimensional subspace. That utilizes a system of idempotent ultrafilters on a families of (partial) semigroups known as $\mathrm{FIN}_k$ (where $k$ ranges over the natural numbers).