{
  "id": "2106.02230",
  "version": "v1",
  "published": "2021-06-04T03:10:21.000Z",
  "updated": "2021-06-04T03:10:21.000Z",
  "title": "Maximal antichains of subsets II: Constructions",
  "authors": [
    "Jerrold R. Griggs",
    "Thomas Kalinowski",
    "Uwe Leck",
    "Ian T. Roberts",
    "Michael Schmitz"
  ],
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "This is the second in a sequence of three papers investigating the question for which positive integers $m$ there exists a maximal antichain of size $m$ in the Boolean lattice $B_n$ (the power set of $[n]:=\\{1,2,\\dots,n\\}$, ordered by inclusion). In the previous paper we characterized those $m$ between the maximum size $\\binom{n}{\\lfloor n/2 \\rfloor}$ and $\\binom{n}{\\lceil n/2\\rceil}-\\lceil n/2\\rceil^2$ that are not sizes of maximal antichains. In this paper we show that all smaller $m$ are sizes of maximal antichains.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-04T03:10:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "06A07",
      "05D05"
    ],
    "keywords": [
      "maximal antichain",
      "constructions",
      "power set",
      "boolean lattice",
      "positive integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}