Higher Dimensional Chain Conditions
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866914913126973440 |
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| author | Todorcevic, Stevo Zhang, Jing |
| author_facet | Todorcevic, Stevo Zhang, Jing |
| contents | We investigate higher dimensional chain conditions, where the largeness notion is given by Fubini products of a given ideal. From strong saturation properties of an ideal, we derive abstractly versions of higher dimensional $Δ$-system lemma, which imply many posets, including any finite support iteration of $σ$-centered posets and measure algebras, satisfy the higher dimensional chain conditions. We then show that if a poset satisfies a strengthening of the $σ$-finite chain condition by Horn and Tarski, then it satisfies higher dimensional chain conditions. As an application, we derive Ramsey-theoretic consequences, namely various partition hypotheses as studied by Bannister, Bergfalk, Moore and Todorcevic, from the existence of ideals satisfying strong chain conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_11369 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Higher Dimensional Chain Conditions Todorcevic, Stevo Zhang, Jing Logic Combinatorics We investigate higher dimensional chain conditions, where the largeness notion is given by Fubini products of a given ideal. From strong saturation properties of an ideal, we derive abstractly versions of higher dimensional $Δ$-system lemma, which imply many posets, including any finite support iteration of $σ$-centered posets and measure algebras, satisfy the higher dimensional chain conditions. We then show that if a poset satisfies a strengthening of the $σ$-finite chain condition by Horn and Tarski, then it satisfies higher dimensional chain conditions. As an application, we derive Ramsey-theoretic consequences, namely various partition hypotheses as studied by Bannister, Bergfalk, Moore and Todorcevic, from the existence of ideals satisfying strong chain conditions. |
| title | Higher Dimensional Chain Conditions |
| topic | Logic Combinatorics |
| url | https://arxiv.org/abs/2310.11369 |