Revisiting $\mathfrak b$ and $\mathfrak d$ through Interval Structures
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911702180691968 |
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| author | Cardona, Miguel A. Marton, Adam |
| author_facet | Cardona, Miguel A. Marton, Adam |
| contents | We investigate a family of relational systems arising from interval partitions of $ω$, inspired by Vojtáš's characterization of the bounding and dominating numbers. By varying the underlying asymptotic quantifiers and interval constraints, we obtain several natural interval-type generalizations.
We show that the universal variants are remarkably robust: in all the discrete, colored, restricted, bounded, and measure-theoretic settings considered here, the associated bounding and dominating numbers coincide with the classical invariants $\mathfrak b$ and $\mathfrak d$. In contrast, the existential variants systematically reverse these invariants, yielding that the bounding number coincides with $\mathfrak d$ and the dominating number coincides with $\mathfrak b$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_21215 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Revisiting $\mathfrak b$ and $\mathfrak d$ through Interval Structures Cardona, Miguel A. Marton, Adam Logic 03E05, 03E15, 03E17, 03E35, 03E40 We investigate a family of relational systems arising from interval partitions of $ω$, inspired by Vojtáš's characterization of the bounding and dominating numbers. By varying the underlying asymptotic quantifiers and interval constraints, we obtain several natural interval-type generalizations. We show that the universal variants are remarkably robust: in all the discrete, colored, restricted, bounded, and measure-theoretic settings considered here, the associated bounding and dominating numbers coincide with the classical invariants $\mathfrak b$ and $\mathfrak d$. In contrast, the existential variants systematically reverse these invariants, yielding that the bounding number coincides with $\mathfrak d$ and the dominating number coincides with $\mathfrak b$. |
| title | Revisiting $\mathfrak b$ and $\mathfrak d$ through Interval Structures |
| topic | Logic 03E05, 03E15, 03E17, 03E35, 03E40 |
| url | https://arxiv.org/abs/2605.21215 |