On cardinal invariants related to Rosenthal families and large-scale topology
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909165131137024 |
|---|---|
| author | Martínez-Celis, Arturo Żuchowski, Tomasz |
| author_facet | Martínez-Celis, Arturo Żuchowski, Tomasz |
| contents | Given a function $f \in ω^ω$, a set $A \in [ω]^ω$ is free for $f$ if $f[A] \cap A$ is finite. For a class of functions $Γ\subseteq ω^ω$, we define $\mathfrak{ros}_Γ$ as the smallest size of a family $\mathcal{A}\subseteq [ω]^ω$ such that for every $f\inΓ$ there is a set $A \in \mathcal{A}$ which is free for $f$, and $Δ_Γ$ as the smallest size of a family $\mathcal{F}\subseteqΓ$ such that for every $A\in[ω]^ω$ there is $f\in\mathcal{F}$ such that $A$ is not free for $f$. We compare several versions of these cardinal invariants with some of the classical cardinal characteristics of the continuum. Using these notions, we partially answer some questions from arXiv:1911.01336 [math.LO] and arXiv:2004.01979 [math.GN]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_06639 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On cardinal invariants related to Rosenthal families and large-scale topology Martínez-Celis, Arturo Żuchowski, Tomasz Logic 03E17, 03E05, 03E35, 03E75 Given a function $f \in ω^ω$, a set $A \in [ω]^ω$ is free for $f$ if $f[A] \cap A$ is finite. For a class of functions $Γ\subseteq ω^ω$, we define $\mathfrak{ros}_Γ$ as the smallest size of a family $\mathcal{A}\subseteq [ω]^ω$ such that for every $f\inΓ$ there is a set $A \in \mathcal{A}$ which is free for $f$, and $Δ_Γ$ as the smallest size of a family $\mathcal{F}\subseteqΓ$ such that for every $A\in[ω]^ω$ there is $f\in\mathcal{F}$ such that $A$ is not free for $f$. We compare several versions of these cardinal invariants with some of the classical cardinal characteristics of the continuum. Using these notions, we partially answer some questions from arXiv:1911.01336 [math.LO] and arXiv:2004.01979 [math.GN]. |
| title | On cardinal invariants related to Rosenthal families and large-scale topology |
| topic | Logic 03E17, 03E05, 03E35, 03E75 |
| url | https://arxiv.org/abs/2404.06639 |