{
  "id": "2411.01782",
  "version": "v1",
  "published": "2024-11-04T04:14:58.000Z",
  "updated": "2024-11-04T04:14:58.000Z",
  "title": "The Turán Density of 4-Uniform Tight Cycles",
  "authors": [
    "Maya Sankar"
  ],
  "comment": "45 pages and a 6-page appendix. 8 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "For any uniformity $r$ and residue $k$ modulo $r$, we give an exact characterization of the $r$-uniform hypergraphs that homomorphically avoid tight cycles of length $k$ modulo $r$, in terms of colorings of $(r-1)$-tuples of vertices. This generalizes the result that a graph avoids all odd closed walks if and only if it is bipartite, as well as a result of Kam\\v cev, Letzter, and Pokrovskiy in uniformity 3. In fact, our characterization applies to a much larger class of families than those of the form $\\mathscr C_k^{(r)}=\\{\\text{$r$-uniform tight cycles of length $k$ modulo $r$}\\}$. We also outline a general strategy to prove that, if $\\mathscr C$ is a family of tight-cycle-like hypergraphs (including but not limited to the families $\\mathscr C_k^{(r)}$) for which the above characterization applies, then all sufficiently long $C\\in \\mathscr C$ will have the same Tur\\'an density. We demonstrate an application of this framework, proving that there exists an integer $L_0$ such that for every $L>L_0$ not divisible by 4, the tight cycle $C^{(4)}_L$ has Tur\\'an density $1/2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-04T04:14:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C65"
    ],
    "keywords": [
      "turán density",
      "turan density",
      "characterization applies",
      "homomorphically avoid tight cycles",
      "exact characterization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}