{
  "id": "1709.04567",
  "version": "v1",
  "published": "2017-09-13T23:48:10.000Z",
  "updated": "2017-09-13T23:48:10.000Z",
  "title": "Definable Combinatorics of Some Borel Equivalence Relations",
  "authors": [
    "William Chan",
    "Connor Meehan"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "If $X$ is a set, $E$ is an equivalence relation on $X$, and $n \\in \\omega$, then define $$[X]^n_E = \\{(x_0, ..., x_{n - 1}) \\in {}^nX : (\\forall i,j)(i \\neq j \\Rightarrow \\neg(x_i \\ E \\ x_j))\\}.$$ For $n \\in \\omega$, a set $X$ has the $n$-J\\'onsson property if and only if for every function $f : [X]^n_= \\rightarrow X$, there exists some $Y \\subseteq X$ with $X$ and $Y$ in bijection so that $f[[Y]^n_=] \\neq X$. A set $X$ has the J\\'onsson property if and only for every function $f : (\\bigcup_{n \\in \\omega}[X]^n_=) \\rightarrow X$, there exists some $Y \\subseteq X$ with $X$ and $Y$ in bijection so that $f[\\bigcup_{n \\in \\omega} [Y]^n_=] \\neq X$. Let $n \\in \\omega$, $X$ be a Polish space, and $E$ be an equivalence relation on $X$. $E$ has the $n$-Mycielski property if and only if for all comeager $C \\subseteq {}^nX$, there is some $\\mathbf{\\Delta_1^1}$ $A \\subseteq X$ so that $E \\leq_{\\mathbf{\\Delta_1^1}} E \\upharpoonright A$ and $[A]^n_E \\subseteq C$. The following equivalence relations will be considered: $E_0$ is defined on ${}^\\omega2$ by $x \\ E_0 \\ y$ if and only if $(\\exists n)(\\forall k > n)(x(k) = y(k))$. $E_1$ is defined on ${}^\\omega({}^\\omega2)$ by $x \\ E_1 \\ y$ if and only if $(\\exists n)(\\forall k > n)(x(k) = y(k))$. $E_2$ is defined on ${}^\\omega2$ by $x \\ E_2 \\ y$ if and only if $\\sum\\{\\frac{1}{n + 1} : n \\in x \\ \\triangle \\ y\\} < \\infty$, where $\\triangle$ denotes the symmetric difference. $E_3$ is defined on ${}^\\omega({}^\\omega2)$ by $x \\ E_3 \\ y$ if and only if $(\\forall n)(x(n) \\ E_0 \\ y(n))$. Holshouser and Jackson have shown that $\\mathbb{R}$ is J\\'onsson under $\\mathsf{AD}$. It will be shown that $E_0$ does not have the $3$-Mycielski property and that $E_1$, $E_2$, and $E_3$ do not have the $2$-Mycielski property. Under $\\mathsf{ZF + AD}$, ${}^\\omega 2 / E_0$ does not have the $3$-J\\'onsson property.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-13T23:48:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "borel equivalence relations",
      "definable combinatorics",
      "mycielski property",
      "jonsson property",
      "symmetric difference"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}