Structures preserved by primitive actions of $S_ω$

Abstract: We present a dichotomy for structures $A$ that are preserved by primitive actions of $S_{\omega} = \text{Sym}({\mathbb N})$: either such a structure interprets all finite structures primitively positively, or it is of a very simple form and in particular has a binary polymorphism $f$ and an automorphism $\alpha$ satisfying $f(x,y) = \alpha(f(y,x))$. It is a consequence of our results that the constraint satisfaction problem for $A$ is in P or NP-complete. To prove our result, we study the first-order reducts of the Johnson graph $J(k)$, for $k \geq 2$, whose automorphism group $G$ equals the action of $S_{\omega}$ on the set $V$ of $k$-element subsets of $\mathbb N$. We use the fact that $J(k)$ has a finitely bounded homogeneous Ramsey expansion and that $G$ is a maximal closed subgroup of $\text{Sym}(V)$.