{
  "id": "2401.16289",
  "version": "v5",
  "published": "2024-01-29T16:49:12.000Z",
  "updated": "2024-02-06T04:43:24.000Z",
  "title": "Turán Densities for Daisies and Hypercubes",
  "authors": [
    "David Ellis",
    "Maria-Romina Ivan",
    "Imre Leader"
  ],
  "comment": "14 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "An $r$-daisy is an $r$-uniform hypergraph consisting of the six $r$-sets formed by taking the union of an $(r-2)$-set with each of the 2-sets of a disjoint 4-set. Bollob\\'as, Leader and Malvenuto, and also Bukh, conjectured that the Tur\\'an density of the $r$-daisy tends to zero as $r \\to \\infty$. In this paper we disprove this conjecture. Adapting our construction, we are also able to disprove a folklore conjecture about Tur\\'an densities of hypercubes. For fixed $d$ and large $n$, we show that the smallest set of vertices of the $n$-dimensional hypercube $Q_n$ that meets every copy of $Q_d$ has asymptotic density strictly below $1/(d+1)$, for all $d \\geq 8$. In fact, we show that this asymptotic density is at most $c^d$, for some constant $c<1$. As a consequence, we obtain similar bounds for the edge-Tur\\'an densities of hypercubes. We also answer some related questions of Johnson and Talbot, and disprove a conjecture made by Bukh and by Griggs and Lu on poset densities.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2024-02-06T04:43:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C65"
    ],
    "keywords": [
      "turán densities",
      "asymptotic density",
      "turan density",
      "dimensional hypercube",
      "smallest set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}