{
  "id": "1711.04353",
  "version": "v1",
  "published": "2017-11-12T20:41:12.000Z",
  "updated": "2017-11-12T20:41:12.000Z",
  "title": "Ordinal Definability and Combinatorics of Equivalence Relations",
  "authors": [
    "William Chan"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "Assume $\\mathsf{ZF + AD^+ + V = L(\\mathscr{P}(\\mathbb{R}))}$. Let $E$ be a $\\mathbf{\\Sigma}^1_1$ equivalence relation coded in $\\mathrm{HOD}$. $E$ has an ordinal definable equivalence class without any ordinal definable elements if and only if $\\mathrm{HOD} \\models E$ is unpinned. $\\mathsf{ZF + AD^+ + V = L(\\mathscr{P}(\\mathbb{R}))}$ proves $E$-class section uniformization when $E$ is a $\\mathbf{\\Sigma}^1_1$ equivalence relation on $\\mathbb{R}$ which is pinned in every transitive model of $\\mathsf{ZFC}$ containing the real which codes $E$: Suppose $R$ is a relation on $\\mathbb{R}$ such that each section $R_x = \\{y : (x,y) \\in R\\}$ is an $E$-class, then there is a function $f : \\mathbb{R} \\rightarrow \\mathbb{R}$ such that for all $x \\in \\mathbb{R}$, $R(x,f(x))$. $\\mathsf{ZF + AD}$ proves that $\\mathbb{R} \\times \\kappa$ is J\\'onsson whenever $\\kappa$ is an ordinal: For every function $f : [\\mathbb{R} \\times \\kappa]^{<\\omega}_= \\rightarrow \\mathbb{R} \\times \\kappa$, there is an $A \\subseteq \\mathbb{R} \\times \\kappa$ with $A$ in bijection with $\\mathbb{R} \\times \\kappa$ and $f[[A]^{<\\omega}_=] \\neq \\mathbb{R} \\times \\kappa$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-12T20:41:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "equivalence relation",
      "ordinal definability",
      "combinatorics",
      "ordinal definable equivalence class",
      "class section uniformization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}