{
  "id": "2207.07401",
  "version": "v1",
  "published": "2022-07-15T11:09:49.000Z",
  "updated": "2022-07-15T11:09:49.000Z",
  "title": "Galvin's property at large cardinals and the axiom of determinancy",
  "authors": [
    "Tom Benhamou",
    "Shimon Garti",
    "Alejandro Poveda"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "In the first part of this paper, we explore the possibility for a very large cardinal $\\kappa$ to carry a $\\kappa$-complete ultrafilter without Galvin's property. In this context, we prove the consistency of every ground model $\\kappa$-complete ultrafilter extends to a non-Galvin one. Oppositely, it is also consistent that every ground model $\\kappa$-complete ultrafilter extends to a $P$-point ultrafilter, hence to another one satisfying Galvin's property. We also study Galvin's property at large cardinals in the choiceless context, especially under \\textsf{AD}. Finally, we apply this property to a classical pro\\-blem in partition calculus by proving the relation $\\lambda\\rightarrow(\\lambda,\\omega+1)^2$ under ``\\textsf{AD}+$V=L(\\mathbb{R})$'' for unboundedly many $\\lambda>{\\rm cf}(\\lambda)>\\omega$ below $\\Theta$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-15T11:09:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E02",
      "03E35",
      "03E60"
    ],
    "keywords": [
      "large cardinal",
      "complete ultrafilter extends",
      "determinancy",
      "ground model",
      "study galvins property"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}