{
  "id": "2109.06140",
  "version": "v1",
  "published": "2021-09-13T17:32:10.000Z",
  "updated": "2021-09-13T17:32:10.000Z",
  "title": "Characterizing the existence of a Borel complete expansion",
  "authors": [
    "Michael C. Laskowski",
    "Douglas S. Ulrich"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "We develop general machinery to cast the class of potential canonical Scott sentences of an infinitary sentence $\\Phi$ as a class of structures in a related language. From this, we show that $\\Phi$ has a Borel complete expansion if and only if $S_\\infty$ divides $Aut(M)$ for some countable model $M\\models \\Phi$. Using this, we prove that for theories $T_h$ asserting that $\\{E_n\\}$ is a countable family of cross cutting equivalence relations with $h(n)$ classes, if $h(n)$ is uniformly bounded then $T_h$ is not Borel complete, providing a converse to Theorem~2.1 of \\cite{LU}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-13T17:32:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C55",
      "03E15"
    ],
    "keywords": [
      "borel complete expansion",
      "cross cutting equivalence relations",
      "potential canonical scott sentences",
      "characterizing",
      "general machinery"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}