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| Main Authors: | , , , |
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| Format: | Preprint |
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2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2206.14936 |
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| _version_ | 1866917199933865984 |
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| author | Brendle, Jörg Guzmán, Osvaldo Hrušák, Michael Raghavan, Dilip |
| author_facet | Brendle, Jörg Guzmán, Osvaldo Hrušák, Michael Raghavan, Dilip |
| contents | We study some strong combinatorial properties of $\textsf{MAD}$ families. An ideal $\mathcal{I}$ is Shelah-Steprāns if for every set $X\subseteq{\left[ ω\right]}^{<ω}$ there is an element of $\mathcal{I}$ that either intersects every set in $X$ or contains infinitely many members of it. We prove that a Borel ideal is Shelah-Steprāns if and only if it is Katětov above the ideal $\textsf{fin}\times\textsf{fin}$. We prove that Shelah-Steprāns $\textsf{MAD}$ families have strong indestructibility properties (in particular, they are both Cohen and random indestructible). We also consider some other strong combinatorial properties of $\textsf{MAD}$ families. Finally, it is proved that it is consistent to have $\mathrm{non}(\mathcal{M}) = {\aleph}_{1}$ and no Shelah-Steprāns families of size ${\aleph}_{1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_14936 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Combinatorial properties of MAD families Brendle, Jörg Guzmán, Osvaldo Hrušák, Michael Raghavan, Dilip Logic General Topology We study some strong combinatorial properties of $\textsf{MAD}$ families. An ideal $\mathcal{I}$ is Shelah-Steprāns if for every set $X\subseteq{\left[ ω\right]}^{<ω}$ there is an element of $\mathcal{I}$ that either intersects every set in $X$ or contains infinitely many members of it. We prove that a Borel ideal is Shelah-Steprāns if and only if it is Katětov above the ideal $\textsf{fin}\times\textsf{fin}$. We prove that Shelah-Steprāns $\textsf{MAD}$ families have strong indestructibility properties (in particular, they are both Cohen and random indestructible). We also consider some other strong combinatorial properties of $\textsf{MAD}$ families. Finally, it is proved that it is consistent to have $\mathrm{non}(\mathcal{M}) = {\aleph}_{1}$ and no Shelah-Steprāns families of size ${\aleph}_{1}$. |
| title | Combinatorial properties of MAD families |
| topic | Logic General Topology |
| url | https://arxiv.org/abs/2206.14936 |