{
  "id": "1602.03690",
  "version": "v1",
  "published": "2016-02-11T12:10:07.000Z",
  "updated": "2016-02-11T12:10:07.000Z",
  "title": "Constructing a weak subset of a random set",
  "authors": [
    "Lu Liu"
  ],
  "comment": "10 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "The tree forcing method given by (Liu 2015) enables the cone avoiding of strong enumeration of a given tree, within a subset or co-subset of an arbitrary given set, provided the given tree does not admit computable strong enumeration. Using this result, we settled and reproduced a series of problems in reverse mathematics. In this paper, we demonstrate cone avoiding results within an infinite subset of a given 1-random set. We show that for any given 1-random set $X$, there exists an infinite subset $Y$ of $X$ such that $Y$ does not compute any real with positive effective Hausdorff dimension, thus answering negatively a question posed by Kjos-Hanssen that whether there exists a 1-random set of which any infinite subset computes some 1-random real. The result is surprising in that the tree forcing technique used on the subset or co-subset seems to heavily rely on subset co-subset combinatorics, whereas this result does not.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-11T12:10:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "68Q30"
    ],
    "keywords": [
      "random set",
      "weak subset",
      "infinite subset",
      "admit computable strong enumeration",
      "demonstrate cone avoiding results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160203690L"
    }
  }
}