Structures preserved by primitive actions of $S_ω$
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911104688455680 |
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| author | Bodirsky, Manuel Bodor, Bertalan |
| author_facet | Bodirsky, Manuel Bodor, Bertalan |
| contents | 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 $\text{Sym}(V)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_03789 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Structures preserved by primitive actions of $S_ω$ Bodirsky, Manuel Bodor, Bertalan Logic Computational Complexity 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 $\text{Sym}(V)$. |
| title | Structures preserved by primitive actions of $S_ω$ |
| topic | Logic Computational Complexity |
| url | https://arxiv.org/abs/2501.03789 |