Structures with not too fast unlabelled growth
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909701519704064 |
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| author | Bodor, Bertalan |
| author_facet | Bodor, Bertalan |
| contents | Let $\mathscr{S}$ be the class of all structures whose growth rate on orbits of subsets of size $n$ is not faster than $\frac{2^n}{p(n)}$ for any polynomial $p$. In this article we give a complete classification of all structures in $\mathscr{S}$ in terms of their automorphism groups. As a consequence of our classification we show that $\mathscr{S}$ has only countably many structures up to bidefinability, all these structures are first-order interpretable in $(\mathbb{Q};<)$ and they are interdefinable with a finitely bounded homogeneous structure. Furthermore, we also show that all structures in $\mathscr{S}$ have finitely many first-order reduct up to interdefinability, thereby confirming Thomas' conjecture for the class $\mathscr{S}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_16985 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Structures with not too fast unlabelled growth Bodor, Bertalan Logic Group Theory 03C52 20B27 20B10 Let $\mathscr{S}$ be the class of all structures whose growth rate on orbits of subsets of size $n$ is not faster than $\frac{2^n}{p(n)}$ for any polynomial $p$. In this article we give a complete classification of all structures in $\mathscr{S}$ in terms of their automorphism groups. As a consequence of our classification we show that $\mathscr{S}$ has only countably many structures up to bidefinability, all these structures are first-order interpretable in $(\mathbb{Q};<)$ and they are interdefinable with a finitely bounded homogeneous structure. Furthermore, we also show that all structures in $\mathscr{S}$ have finitely many first-order reduct up to interdefinability, thereby confirming Thomas' conjecture for the class $\mathscr{S}$. |
| title | Structures with not too fast unlabelled growth |
| topic | Logic Group Theory 03C52 20B27 20B10 |
| url | https://arxiv.org/abs/2507.16985 |