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arXiv · 2404.06639

On cardinal invariants related to Rosenthal families and large-scale topology

Arturo Martínez-Celis·Tomasz Żuchowski
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Abstract

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].

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Arturo Martínez-Celis, Tomasz Żuchowski. 2024-04-09. On cardinal invariants related to Rosenthal families and large-scale topology. https://arxiv.org/abs/2404.06639

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