{
  "id": "1912.10332",
  "version": "v1",
  "published": "2019-12-21T20:40:05.000Z",
  "updated": "2019-12-21T20:40:05.000Z",
  "title": "Selective independence",
  "authors": [
    "Vera Fischer"
  ],
  "comment": "8 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "Let $\\mathfrak{i}$ denote the minimal cardinality of a maximal independent family and let $\\mathfrak{a}_T$ denote the minimal cardinality of a maximal family of pairwise almost disjoint subtrees of $2^{<\\omega}$. Using a countable support iteration of proper, $^\\omega\\omega$-bounding posets of length $\\omega_2$ over a model of CH, we show that consistently $\\mathfrak{i}<\\mathfrak{a}_T$. Moreover, we show that the inequality can be witnessed by a co-analytic maximal independent family of size $\\aleph_1$ in the presence of a $\\Delta^1_3$ definable well-order of the reals. The main result of the paper can be viewed as a partial answer towards the well-known open problem of the consistency of $\\mathfrak{i}<\\mathfrak{a}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-21T20:40:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E17",
      "03E35",
      "03E15"
    ],
    "keywords": [
      "selective independence",
      "minimal cardinality",
      "well-known open problem",
      "disjoint subtrees",
      "countable support iteration"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}