  We prove that any strongly mixing action of a countable abelian group on a
probability space has higher order mixing properties. This is achieved via
introducing and utilizing $\mathcal R$-limits, a notion of convergence which is
based on the classical Ramsey Theorem. $\mathcal R$-limits are intrinsically
connected with a new combinatorial notion of largeness which is similar to but
has stronger properties than the classical notions of uniform density one and
IP$^*$. While the main goal of this paper is to establish a
$\textit{universal}$ property of strongly mixing actions of countable abelian
groups, our results, when applied to $\mathbb Z$-actions, offer a new way of
dealing with strongly mixing transformations. In particular, we obtain several
new characterizations of strong mixing for $\mathbb Z$-actions, including a
result which can be viewed as the analogue of the weak mixing of all orders
property established by Furstenberg in the course of his proof of Szemer\'edi's
theorem. We also demonstrate the versatility of $\mathcal R$-limits by
obtaining new characterizations of higher order weak and mild mixing for
actions of countable abelian groups.
