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arXiv · 2610.28275

Closed maximal eventually different families for $\boldsymbolΣ^0_3$ and weakly Farah ideals

Jialiang He·Jiaheng Zhuang·Yi Zhou
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Abstract

Motivated by the existence of closed maximal $\mathcal{I}$-eventually different ($\mathcal{I}$-MED) families, we study uniform combinatorial properties of ideals. For a fixed ideal, a property is uniform if its witnesses can be chosen by a single Borel map. We prove that every $\boldsymbolΣ^0_3$ ideal is uniformly weakly $P^+$. We also prove that the classes of Farah and weakly Farah ideals coincide, answering a question of Hrušák and Meza-Alcántara. We then show that every weakly Farah ideal is uniformly weakly Ramsey. As an application, we obtain closed $\mathcal{I}$-MED families for $\boldsymbolΣ^0_3$ ideals and weakly Farah ideals.

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Jialiang He, Jiaheng Zhuang, Yi Zhou. 2026-10-20. Closed maximal eventually different families for $\boldsymbolΣ^0_3$ and weakly Farah ideals. https://arxiv.org/abs/2610.28275

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