{
  "id": "2406.11999",
  "version": "v1",
  "published": "2024-06-17T18:12:54.000Z",
  "updated": "2024-06-17T18:12:54.000Z",
  "title": "Supersaturation of Tree Posets",
  "authors": [
    "Tao Jiang",
    "Sean Longbrake",
    "Sam Spiro",
    "Liana Yepremyan"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A well-known result of Bukh shows that if $P$ is a tree poset of height $k$, then any subset of the Boolean lattice $\\mathcal{F}\\subseteq \\mathcal{B}_n$ of size at least $(k-1+\\varepsilon){n\\choose \\lceil n/2\\rceil}$ contains at least one copy of $P$. This was extended by Boehnlein and Jiang to induced copies. We strengthen both results by showing that for any integer $q\\ge k$, any family $\\mathcal{F}$ of size at least $(q-1+\\varepsilon){n\\choose \\lceil n/2\\rceil}$ contains on the order of as many induced copies of $P$ as is contained in the $q$ middle layers of the Boolean lattice. This answers a conjecture of Gerbner, Nagy, Patk\\'os, and Vizer in a strong form for tree posets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-17T18:12:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D05"
    ],
    "keywords": [
      "tree poset",
      "supersaturation",
      "boolean lattice",
      "induced copies",
      "well-known result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}