{
  "id": "1601.01012",
  "version": "v1",
  "published": "2016-01-05T23:00:14.000Z",
  "updated": "2016-01-05T23:00:14.000Z",
  "title": "Perfect Set Theorems for Equivalence Relations with $I$ - small classes",
  "authors": [
    "Ohad Drucker"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "A classical theorem due to Mycielski states that an equivalence relation $E$ having the Baire property and meager equivalence classes must have a perfect set of pairwise inequivalent elements. We consider equivalence relations with $I$ - small equivalence classes, where $I$ is a proper $\\sigma$ - ideal, and ask whether they have a perfect set of pairwise inequivalent elements. We give a positive answer for $E$ universally Baire. We show that the answer for $E$ $\\mathbf{\\Delta_{2}^{1}}$ is independent of $ZFC$, and find set theoretic assumptions equivalent to it when $I$ is the countable ideal. For equivalence relations which are $\\mathbf{\\Sigma^1_2}$ and with meager classes, we show that a perfect set of pairwise inequivalent elements exists whenever a Cohen real over $L[z]$ exists for any real $z$ - which strengthens Mycielski's theorem. A few comments are made about $\\sigma$ - ideals generated by $\\Pi_{1}^{1}$ and orbit equivalence relations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-05T23:00:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "perfect set theorems",
      "small classes",
      "pairwise inequivalent elements",
      "set theoretic assumptions equivalent",
      "strengthens mycielskis theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160101012D"
    }
  }
}