The Polynomial Hierarchy and $\omega$-categorical CSPs

In 2008, Bodirsky and Grohe showed that for every $\Pi_n^{\mathrm{P}}$-level of the Polynomial Hierarchy (PH) there are $\omega$-categorical Constraint Satisfaction Problems (CSPs) complete for this level. We show that, in fact, there are $\omega$-categorical CSPs complete for any level of the PH. To this end, we use a recent result of Bodirsky, Kn\"{a}uer, and Rudolph for constructing $\omega$-categorical CSPs from sentences of Monadic Second-Order logic (MSO) with certain preservation properties. As a secondary contribution, we develop a new tool for producing MSO sentences satisfying said preservation properties.