{
  "id": "2210.08370",
  "version": "v1",
  "published": "2022-10-15T20:41:03.000Z",
  "updated": "2022-10-15T20:41:03.000Z",
  "title": "A Proof of the $(n,k,t)$ Conjectures",
  "authors": [
    "Stacie Baumann",
    "Joseph Briggs"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "An $(n,k,t)$-graph is a graph on $n$ vertices in which every set of $k$ vertices contain a clique on $t$ vertices. Tur\\'an's Theorem (complemented) states that the unique minimum $(n,k,2)$-graph is a disjoint union of cliques. We prove that minimum $(n,k,t)$-graphs are always disjoint unions of cliques for any $t$ (despite nonuniqueness of extremal examples), thereby generalizing Tur\\'an's Theorem and confirming two conjectures of Hoffman et al.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-15T20:41:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C35"
    ],
    "keywords": [
      "conjectures",
      "disjoint union",
      "unique minimum",
      "generalizing turans theorem",
      "despite nonuniqueness"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}