{
  "id": "2406.15263",
  "version": "v1",
  "published": "2024-06-21T15:54:09.000Z",
  "updated": "2024-06-21T15:54:09.000Z",
  "title": "On Stability and Existence of Models in Abstract Elementary Classes",
  "authors": [
    "Marcos Mazari-Armida",
    "Sebastien Vasey",
    "Wentao Yang"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "For an abstract elementary class $\\mathbf{K}$ and a cardinal $\\lambda \\geq LS(\\mathbf{K})$, we prove under mild cardinal arithmetic assumptions, categoricity in two succesive cardinals, almost stability for $\\lambda^+$-minimal types and continuity of splitting in $\\lambda$, that stability in $\\lambda$ is equivalent to the existence of a model in $\\lambda^{++}$. The forward direction holds without any cardinal or categoricity assumptions, this result improves both [Vas18b, 12.1] and [MaYa24, 3.14]. Moreover, we prove a categoricity theorem for abstract elementary classes with weak amalgamation and tameness under mild structural assumptions in $\\lambda$. A key feature of this result is that we do not assume amalgamation or arbitrarily large models.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-21T15:54:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C48",
      "03C45",
      "03C52",
      "03C55"
    ],
    "keywords": [
      "abstract elementary class",
      "mild cardinal arithmetic assumptions",
      "categoricity",
      "mild structural assumptions",
      "large models"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}