{
  "id": "1811.07210",
  "version": "v2",
  "published": "2018-11-17T19:22:47.000Z",
  "updated": "2018-12-06T12:16:30.000Z",
  "title": "Vaught's Conjecture for Monomorphic Theories",
  "authors": [
    "Miloš S. Kurilić"
  ],
  "comment": "13 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "A complete first order theory of a relational signature is called monomorphic iff all its models are monomorphic (i.e. have all the $n$-element substructures isomorphic, for each positive integer $n$). We show that a complete theory ${\\mathcal T}$ having infinite models is monomorphic iff it has a countable monomorphic model and confirm the Vaught conjecture for monomorphic theories. More precisely, we prove that if ${\\mathcal T}$ is a complete monomorphic theory having infinite models, then the number of its non-isomorphic countable models, $I({\\mathcal T},\\omega )$, is either equal to $1$ or to ${\\mathfrak c}$. In addition, $I({\\mathcal T},\\omega)= 1$ iff some countable model of ${\\mathcal T}$ is simply definable by an $\\omega$-categorical linear order on its domain.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2018-12-06T12:16:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C15"
    ],
    "keywords": [
      "vaughts conjecture",
      "complete first order theory",
      "infinite models",
      "element substructures isomorphic",
      "complete monomorphic theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}