{
  "id": "2404.06639",
  "version": "v1",
  "published": "2024-04-09T22:24:32.000Z",
  "updated": "2024-04-09T22:24:32.000Z",
  "title": "On cardinal invariants related to Rosenthal families and large-scale topology",
  "authors": [
    "Arturo Martínez-Celis",
    "Tomasz Żuchowski"
  ],
  "comment": "19 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "Given a function $f \\in \\omega^\\omega$, a set $A \\in [\\omega]^\\omega$ is free for $f$ if $f[A] \\cap A$ is finite. For a class of functions $\\Gamma \\subseteq \\omega^{\\omega}$, we define $\\mathfrak{ros}_\\Gamma$ as the smallest size of a family $\\mathcal{A}\\subseteq [\\omega]^\\omega$ such that for every $f\\in\\Gamma$ there is a set $A \\in \\mathcal{A}$ which is free for $f$, and $\\Delta_\\Gamma$ as the smallest size of a family $\\mathcal{F}\\subseteq\\Gamma$ such that for every $A\\in[\\omega]^\\omega$ 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].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-09T22:24:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E17",
      "03E05",
      "03E35",
      "03E75"
    ],
    "keywords": [
      "cardinal invariants",
      "rosenthal families",
      "large-scale topology",
      "smallest size",
      "classical cardinal characteristics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}