← 研踪 PaperTrailarXiv:2501.03789 · v5
Structures preserved by primitive actions of $S_ω$
首次提交 2025-01-07 21:54
版本更新 2026-11-01 23:41
雷达观测 2026-11-04 23:43
摘要
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 $\text{Sym}(V)$.
作者备注：34 pages plus references; accepted for publication in LMCS
观测到的版本
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v5 · 2026-11-01 23:41 · 最新观测版本
Structures preserved by primitive actions of $S_ω$
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 $\text{Sym}(V)$.
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