{
  "id": "2209.11226",
  "version": "v1",
  "published": "2022-09-22T17:59:36.000Z",
  "updated": "2022-09-22T17:59:36.000Z",
  "title": "The Halpern--Läuchli Theorem at singular cardinals and failures of weak versions",
  "authors": [
    "Natasha Dobrinen",
    "Saharon Shelah"
  ],
  "comment": "15 pages. This is paper 1230 on Shelah's list",
  "categories": [
    "math.LO",
    "math.CO"
  ],
  "abstract": "This paper continues a line of investigation of the Halpern--L\\\"{a}uchli Theorem at uncountable cardinals. We prove in ZFC that the Halpern--L\\\"{a}uchli Theorem for one tree of height $\\kappa$ holds whenever $\\kappa$ is strongly inaccessible and the coloring takes less than $\\kappa$ colors. We prove consistency of the Halpern--L\\\"{a}uchli Theorem for finitely many trees of height $\\kappa$, where $\\kappa$ is a strong limit cardinal of countable cofinality. On the other hand, we prove failure of a weak form of Halpern--\\Lauchli\\ for trees of height $\\kappa$, whenever $\\kappa$ is a strongly inaccessible, non-Mahlo cardinal or a singular strong limit cardinal with cofinality the successor of a regular cardinal. We also prove failure in $L$ of an intermediate version for all strongly inaccessible, non-weakly compact cardinals.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-22T17:59:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E02",
      "03E05",
      "03E10",
      "03E35",
      "05D10"
    ],
    "keywords": [
      "singular cardinals",
      "weak versions",
      "halpern-läuchli theorem",
      "singular strong limit cardinal",
      "strongly inaccessible"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}