Higher Dimensional Chain Conditions

Stevo Todorcevic; Jing Zhang

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arXiv · 2310.11369

Higher Dimensional Chain Conditions

Stevo Todorcevic·Jing Zhang

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Abstract

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 $\Delta$-system lemma, which imply many posets, including any finite support iteration of $\sigma$-centered posets and measure algebras, satisfy the higher dimensional chain conditions. We then show that if a poset satisfies a strengthening of the $\sigma$-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.

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BibTeX RIS

Stevo Todorcevic, Jing Zhang. 2023-10-17. Higher Dimensional Chain Conditions. https://arxiv.org/abs/2310.11369

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