{
  "id": "1907.03142",
  "version": "v1",
  "published": "2019-07-06T15:44:25.000Z",
  "updated": "2019-07-06T15:44:25.000Z",
  "title": "Indestructibility of the tree property",
  "authors": [
    "Radek Honzik",
    "Sarka Stejskalova"
  ],
  "comment": "22 pages, submitted",
  "categories": [
    "math.LO"
  ],
  "abstract": "In the first part of the paper, we show that if $\\omega \\le \\kappa < \\lambda$ are cardinals, $\\kappa^{<\\kappa} = \\kappa$, and $\\lambda$ is weakly compact, then in $V[\\M(\\kappa,\\lambda)]$ the tree property at $\\lambda = \\kappa^{++V[\\M(\\kappa,\\lambda)]}$ is indestructible under all $\\kappa^+$-cc forcing notions which live in $V[\\Add(\\kappa,\\lambda)]$, where $\\Add(\\kappa,\\lambda)$ is the Cohen forcing for adding $\\lambda$-many subsets of $\\kappa$ and $\\M(\\kappa,\\lambda)$ is the standard Mitchell forcing for obtaining the tree property at $\\lambda = (\\kappa^{++})^{V[\\M(\\kappa,\\lambda)]}$. This result has direct applications to Prikry-type forcing notions and generalized cardinal invariants. In the second part, we assume that $\\lambda$ is supercompact and generalize the construction and obtain a model $V^*$, a generic extension of $V$, in which the tree property at $(\\kappa^{++})^{V^*}$ is indestructible under all $\\kappa^+$-cc forcing notions living in $V[\\Add(\\kappa,\\lambda)]$, and in addition by all forcing notions living in $V^*$ which are $\\kappa^+$-closed and ``liftable'' in a prescribed sense (such as $\\kappa^{++}$-directed closed forcings or well-met forcings which are $\\kappa^{++}$-closed with the greatest lower bounds).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-06T15:44:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tree property",
      "cc forcing notions",
      "indestructibility",
      "greatest lower bounds",
      "forcing notions living"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}