Structures preserved by primitive actions of $S_{\omega }$

We present a dichotomy for structures $A$ that are preserved by primitive actions of $S_{\omega }=\text{Sym}(\mathbb{N})$: such a structure primitively positively constructs all finite structures and the constraint satisfaction problem is NP-complete, or the constraint satisfaction problem for $A$ is in P. To prove our result, we study the first-order reducts of the Johnson graph $J(k)$, for $k\ge 2$, whose automorphism group $G$ equals the action of $\text{Sym}(\mathbb{N})$ 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 $\text{Sym}(V)$.