{
  "id": "2011.01273",
  "version": "v1",
  "published": "2020-11-02T19:31:28.000Z",
  "updated": "2020-11-02T19:31:28.000Z",
  "title": "The independence of GCH and a combinatorial principle related to Banach-Mazur games",
  "authors": [
    "Will Brian",
    "Alan Dow",
    "Saharon Shelah"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "It was proved recently that Telg\\'arsky's conjecture, which concerns partial information strategies in the Banach-Mazur game, fails in models of $\\mathsf{GCH}+\\square$. The proof introduces a combinatorial principle that is shown to follow from $\\mathsf{GCH}+\\square$, namely: $\\triangledown$: Every separative poset $\\mathbb P$ with the $\\kappa$-cc contains a dense sub-poset $\\mathbb D$ such that $|\\{ q \\in \\mathbb D \\,:\\, p \\text{ extends } q \\}| < \\kappa$ for every $p \\in \\mathbb P$. We prove this principle is independent of $\\mathsf{GCH}$ and $\\mathsf{CH}$, in the sense that $\\triangledown$ does not imply $\\mathsf{CH}$, and $\\mathsf{GCH}$ does not imply $\\triangledown$ assuming the consistency of a huge cardinal. We also consider the more specific question of whether $\\triangledown$ holds with $\\mathbb P$ equal to the weight-$\\aleph_\\omega$ measure algebra. We prove, again assuming the consistency of a huge cardinal, that the answer to this question is independent of $\\mathsf{ZFC}+\\mathsf{GCH}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-02T19:31:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "banach-mazur game",
      "combinatorial principle",
      "independence",
      "concerns partial information strategies",
      "huge cardinal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}