{
  "id": "2402.14113",
  "version": "v3",
  "published": "2024-02-21T20:28:37.000Z",
  "updated": "2024-06-05T18:11:04.000Z",
  "title": "Saturation of $k$-chains in the Boolean lattice",
  "authors": [
    "Ryan R. Martin",
    "Nick Veldt"
  ],
  "comment": "11 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Given a set $X$, a collection $\\mathcal{F} \\subset \\mathcal{P}(X)$ is said to be $k$-Sperner if it does not contain a chain of length $k+1$ under set inclusion and it is saturated if it is maximal with respect to this probability. Gerbner et al. proved that the smallest saturated $k$-Sperner system contains at least $2^{k/2-1}$ elements, and later, Morrison, Noel, and Scott showed that the smallest such set contains no more than $2^{0.976723k}$ elements. We improve both the upper and lower bounds, showing that the size of the smallest saturated $k$-Sperner system lies between $\\sqrt{k}2^{k/2}$ and $2^{0.961471k}$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2024-06-05T18:11:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "boolean lattice",
      "saturation",
      "sperner system contains",
      "sperner system lies",
      "set inclusion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}