{
  "id": "1906.00843",
  "version": "v1",
  "published": "2019-06-03T14:41:26.000Z",
  "updated": "2019-06-03T14:41:26.000Z",
  "title": "Towers and gaps at uncountable cardinals",
  "authors": [
    "Vera Fischer",
    "Diana Carolina Montoya",
    "Jonathan Schilhan",
    "Dániel T. Soukup"
  ],
  "comment": "24 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "Our goal is to study the pseudo-intersection and tower numbers on uncountable regular cardinals, whether these two cardinal characteristics are necessarily equal, and related problems on the existence of gaps. First, we prove that either $\\mathfrak p(\\kappa)=\\mathfrak t(\\kappa)$ or there is a $(\\mathfrak p(\\kappa),\\lambda)$-gap of club-supported slaloms for some $\\lambda< \\mathfrak p(\\kappa)$. While the existence of such gaps is unclear, this is a promising step to lift Malliaris and Shelah's proof of $\\mathfrak p=\\mathfrak t$ to uncountable cardinals. We do analyze gaps of slaloms and, in particular, show that $\\mathfrak p(\\kappa)$ is always regular; the latter extends results of Garti. Finally, we turn to club variants of $\\mathfrak p(\\kappa)$ and present a new model for the inequality $\\mathfrak{p}(\\kappa) = \\kappa^+ < \\mathfrak{p}_{cl}(\\kappa) = 2^\\kappa$. In contrast to earlier arguments by Shelah and Spasojevic, we achieve this by adding $\\kappa$-Cohen reals and then successively diagonalising the club-filter; the latter is shown to preserve a Cohen witness to $\\mathfrak{p}(\\kappa) = \\kappa^+$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-03T14:41:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E05",
      "03E17"
    ],
    "keywords": [
      "uncountable cardinals",
      "cohen witness",
      "cardinal characteristics",
      "tower numbers",
      "uncountable regular cardinals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}