We present a dichotomy for structures $A$ that are preserved by primitive actions of $S_ω = \text{Sym}({\mathbb N})$: 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 $k \geq 2$, 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 $\text{Sym}(V)$.

翻译：我们针对由$S_ω = \text{Sym}({\mathbb N})$的原始作用所保持的结构$A$提出一个二分法：此类结构要么原始地积极构造所有有限结构且其约束满足问题是NP完全的，要么结构$A$的约束满足问题属于P类。为证明此结果，我们研究了Johnson图$J(k)$（其中$k \geq 2$）的一阶归约，其自同构群$G$等于$\text{Sym}({\mathbb N})$在${\mathbb N}$的$k$元子集构成的集合$V$上的作用。我们利用了$J(k)$具有有限有界齐次Ramsey扩张这一事实，以及$G$是$\text{Sym}(V)$的极大闭子群这一性质。