The CSP Dichotomy, the Axiom of Choice, and Cyclic Polymorphisms

We study Constraint Satisfaction Problems (CSPs) in an infinite context. We show that the dichotomy between easy and hard problems – established already in the finite case – presents itself as the strength of the corresponding De Bruijin-Erdős-type compactness theorem over ZF. More precisely, if $\mathcal{D}$ is a structure, let $K_{\mathcal{D}}$ stand for the following statement: for every structure $\mathcal{X}$ if every finite substructure of $\mathcal{X}$ admits a solution to $\mathcal{D}$, then so does $\mathcal{X}$. We prove that if $\mathcal{D}$ admits no cyclic polymorphism, and thus it is NP-complete by the CSP Dichotomy Theorem, then $K_{\mathcal{D}}$ is equivalent to the Boolean Prime Ideal Theorem (BPI) over ZF. Conversely, we also show that if $\mathcal{D}$ admits a cyclic polymorphism, and thus it is in P, then $K_{\mathcal{D}}$ is strictly weaker than BPI.