Indiscernible extraction at small large cardinals from a higher-arity stability notion
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912459839766528 |
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| author | Hanson, James E. |
| author_facet | Hanson, James E. |
| contents | We introduce a higher-arity stability notion defined in terms of $k$-splitting, a higher-arity generalization of splitting. We show that theories with bounded $k$-splitting have improved indiscernible extraction at $k$-ineffable cardinals, and we give a non-trivial example of a theory with bounded $k$-splitting but unbounded $(k-1)$-splitting for each odd $k > 1$. We also show that bounded $k$-splitting implies $\mathrm{NFOP}_k$, a higher arity stability notion introduced by Terry and Wolf. We then use our indiscernible extraction result together with a construction of Kaplan and Shelah to give a strong counterexample to the converse: an $\mathrm{NIP}$ theory with unbounded $k$-splitting for every $k$. Finally, as a thematically related but technically independent result, we show that treelessness implies $\mathrm{NFOP}_2$, sharpening a result of Kaplan, Ramsey, and Simon. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_19147 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Indiscernible extraction at small large cardinals from a higher-arity stability notion Hanson, James E. Logic 03C45 We introduce a higher-arity stability notion defined in terms of $k$-splitting, a higher-arity generalization of splitting. We show that theories with bounded $k$-splitting have improved indiscernible extraction at $k$-ineffable cardinals, and we give a non-trivial example of a theory with bounded $k$-splitting but unbounded $(k-1)$-splitting for each odd $k > 1$. We also show that bounded $k$-splitting implies $\mathrm{NFOP}_k$, a higher arity stability notion introduced by Terry and Wolf. We then use our indiscernible extraction result together with a construction of Kaplan and Shelah to give a strong counterexample to the converse: an $\mathrm{NIP}$ theory with unbounded $k$-splitting for every $k$. Finally, as a thematically related but technically independent result, we show that treelessness implies $\mathrm{NFOP}_2$, sharpening a result of Kaplan, Ramsey, and Simon. |
| title | Indiscernible extraction at small large cardinals from a higher-arity stability notion |
| topic | Logic 03C45 |
| url | https://arxiv.org/abs/2506.19147 |