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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.04934 |
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| _version_ | 1866910194563284992 |
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| author | Usvyatsov, Alexander |
| author_facet | Usvyatsov, Alexander |
| contents | We prove that in a countable theory $T$ fully stable over a predicate $P$, any $\lam$-complete set $A$ has the $\lam$-existence property. This means that $A$ can be extended to a $\lam$-saturated model of $T$ without changing the $P$-part. The notion of $\lam$-completeness, introduced in this paper, captures some obvious necessary conditions for such an extension to be possible (for example, the $P$-part of $A$ has to be a $\lam$-saturated model of the appropriate theory). So in a fully stable theory $T$, $\lam$-existence can only fail for trivial reasons. This generalizes results of Chatzidakis in the context of difference fields of characteristic 0. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_04934 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On $\lam$-existence over a predicate Usvyatsov, Alexander Logic We prove that in a countable theory $T$ fully stable over a predicate $P$, any $\lam$-complete set $A$ has the $\lam$-existence property. This means that $A$ can be extended to a $\lam$-saturated model of $T$ without changing the $P$-part. The notion of $\lam$-completeness, introduced in this paper, captures some obvious necessary conditions for such an extension to be possible (for example, the $P$-part of $A$ has to be a $\lam$-saturated model of the appropriate theory). So in a fully stable theory $T$, $\lam$-existence can only fail for trivial reasons. This generalizes results of Chatzidakis in the context of difference fields of characteristic 0. |
| title | On $\lam$-existence over a predicate |
| topic | Logic |
| url | https://arxiv.org/abs/2605.04934 |