{
  "id": "1805.06767",
  "version": "v1",
  "published": "2018-05-17T13:43:58.000Z",
  "updated": "2018-05-17T13:43:58.000Z",
  "title": "Model theory of Steiner triple systems",
  "authors": [
    "Silvia Barbina",
    "Enrique Casanovas"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "A Steiner triple system is a set $S$ together with a collection $\\mathcal{B}$ of subsets of $S$ of size 3 such that any two elements of $S$ belong to exactly one element of $\\mathcal{B}$. It is well known that the class of finite Steiner triple systems has a Fra\\\"{\\i}ss\\'e limit $M_{\\mathrm{F}}$. Here we show that the theory $T^\\ast_\\mathrm{Sq}$ of $M_{\\mathrm{F}}$ is the model completion of the theory of Steiner triple systems. We also prove that $T^\\ast_\\mathrm{Sq}$ has quantifier elimination, it is not small and has $\\mathrm{TP}_2$ and $\\mathrm{NSOP}_1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-17T13:43:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "model theory",
      "finite steiner triple systems",
      "quantifier elimination",
      "model completion",
      "collection"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}