{
  "id": "2303.10530",
  "version": "v1",
  "published": "2023-03-19T01:56:02.000Z",
  "updated": "2023-03-19T01:56:02.000Z",
  "title": "Turán density of long tight cycle minus one hyperedge",
  "authors": [
    "József Balogh",
    "Haoran Luo"
  ],
  "comment": "23 pages, 3 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Denote by $\\mathcal{C}^-_{\\ell}$ the $3$-uniform hypergraph obtained by removing one hyperedge from the tight cycle on $\\ell$ vertices. It is conjectured that the Tur\\'an density of $\\mathcal{C}^-_{5}$ is $1/4$. In this paper, we make progress toward this conjecture by proving that the Tur\\'an density of $\\mathcal{C}^-_{\\ell}$ is $1/4$, for every sufficiently large $\\ell$ not divisible by $3$. One of the main ingredients of our proof is a forbidden-subhypergraph characterization of the hypergraphs, for which there exists a tournament on the same vertex set such that every hyperedge is a cyclic triangle in this tournament. A byproduct of our method is a human-checkable proof for the upper bound on the maximum number of almost similar triangles in a planar point set, which recently was proved using flag algebra by Balogh, Clemen, and Lidick\\'y.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-19T01:56:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C35",
      "05C65",
      "05D05"
    ],
    "keywords": [
      "long tight cycle minus",
      "turán density",
      "turan density",
      "planar point set",
      "uniform hypergraph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}