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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2507.06977 |
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| _version_ | 1866911047701495808 |
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| author | Day, Gabriel |
| author_facet | Day, Gabriel |
| contents | In this paper, we introduce novel variations on several well-known model-theoretic tree properties, and prove several equivalences to known properties. Motivated by the study of generalized indiscernibles, we introduce the notion of the $\calI$-tree property ($\calI$-TP), for an arbitrary Ramsey index structure $\calI$. We focus attention on the colored linear order index structure \textbf{c}, showing that \textbf{c}-TP is equivalent to instability. After introducing \textbf{c}-$\TPi$ and \textbf{c}-$\TPii$, we prove that \textbf{c}-$\TPi$ is equivalent to $\TPi$, and that \textbf{c}-$\TPii$ is equivalent to IP. We see that these three tree properties give a dichotomy theorem, just as with TP, $\TPi$, and $\TPii$. Along the way, we observe that appropriately generalized tree index structures $\calI^{<ω}$ are Ramsey, allowing for the use of generalized tree indiscernibles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_06977 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Results on Colored Tree Properties Day, Gabriel Logic In this paper, we introduce novel variations on several well-known model-theoretic tree properties, and prove several equivalences to known properties. Motivated by the study of generalized indiscernibles, we introduce the notion of the $\calI$-tree property ($\calI$-TP), for an arbitrary Ramsey index structure $\calI$. We focus attention on the colored linear order index structure \textbf{c}, showing that \textbf{c}-TP is equivalent to instability. After introducing \textbf{c}-$\TPi$ and \textbf{c}-$\TPii$, we prove that \textbf{c}-$\TPi$ is equivalent to $\TPi$, and that \textbf{c}-$\TPii$ is equivalent to IP. We see that these three tree properties give a dichotomy theorem, just as with TP, $\TPi$, and $\TPii$. Along the way, we observe that appropriately generalized tree index structures $\calI^{<ω}$ are Ramsey, allowing for the use of generalized tree indiscernibles. |
| title | Results on Colored Tree Properties |
| topic | Logic |
| url | https://arxiv.org/abs/2507.06977 |