发表机构
Institute of Discrete Mathematics and Geometry, TU Wien（维也纳工业大学离散数学与几何研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究普遍空集上的基数不变量，证明了多个相关不变量之间的不等式关系，并探讨了在特定假设下的一致性结果。
AI 中文摘要
我们研究普遍空集上的基数不变量。特别是，我们证明在ZFC中，$\mathfrak{b} < \operatorname{cof}(\mathcal{UN})$ 和 $\operatorname{non}(\mathcal{N}) = \operatorname{non}(\mathcal{UN}) < \operatorname{cof}(\mathcal{UN})$。此外，假设 $\operatorname{add}(\mathcal{N}) = \mathfrak{c}$，我们通过改编Yorioka的技术证明 $\operatorname{cof}(\mathcal{UN}) = \mathfrak{d}_\mathfrak{c}$。此外，我们证明 $\operatorname{add}(\mathcal{UN}) < \operatorname{cov}(\mathcal{UN}) < \operatorname{non}(\mathcal{UN}) < \operatorname{cof}(\mathcal{UN})$ 的一致性。
英文摘要
We study universally null sets in forcing extensions and the cardinal invariants of their ideal $\mathcal{UN}$. We prove that the Miller model contains a nonmeager universally null set of cardinality $\aleph_2=\mathfrak{c}$, answering a question of Brendle and Larson. The proof uses preservation of universal nullity by countable support iterations of Miller forcing and shows that the set of Miller reals added during the iteration is universally null. We also prove that every universally null set in the $\mathbb{PT}_{f,g}$ model has cardinality at most $\aleph_1$, and give a consistent example of a universally null set that acquires a perfect subset after an $ω_1$-preserving forcing. For the cardinal invariants, we show that $|X|<\operatorname{cof}(\mathcal{UN})$ for every $X\in\mathcal{UN}$, and that $\operatorname{add}(\mathcal{N})=\mathfrak{c}$ implies $\operatorname{cof}(\mathcal{UN})=\mathfrak{d}_{\mathfrak{c}}$. Finally, we prove the consistency of $\operatorname{add}(\mathcal{UN})<\operatorname{cov}(\mathcal{UN}) <\operatorname{non}(\mathcal{UN})<\operatorname{cof}(\mathcal{UN})$.
