{
  "id": "2009.06445",
  "version": "v1",
  "published": "2020-09-14T13:59:14.000Z",
  "updated": "2020-09-14T13:59:14.000Z",
  "title": "Tree forcing and definable maximal independent sets in hypergraphs",
  "authors": [
    "Jonathan Schilhan"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "We show that after forcing with a countable support iteration or a finite product of Sacks or splitting forcing over $L$, every analytic hypergraph on a Polish space admits a $\\mathbf{\\Delta}^1_2$ maximal independent set. As a main application we get the consistency of $\\mathfrak{r} = \\mathfrak{u} = \\mathfrak{i} = \\omega_2$ together with the existence of a $\\Delta^1_2$ ultrafilter, a $\\Pi^1_1$ maximal independent family and a $\\Delta^1_2$ Hamel basis. This solves open problems of Brendle, Fischer and Khomskii and the author. We also show in ZFC that $\\mathfrak{d} \\leq \\mathfrak{i}_{cl}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-14T13:59:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E35",
      "03E17",
      "03E15"
    ],
    "keywords": [
      "definable maximal independent sets",
      "tree forcing",
      "polish space admits",
      "countable support iteration",
      "open problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}