{
  "id": "2205.07392",
  "version": "v1",
  "published": "2022-05-15T22:39:42.000Z",
  "updated": "2022-05-15T22:39:42.000Z",
  "title": "Saturation for Small Antichains",
  "authors": [
    "Irina Đanković",
    "Maria-Romina Ivan"
  ],
  "comment": "8 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "For a given positive integer $k$ we say that a family of subsets of $[n]$ is $k$-antichain saturated if it does not contain $k$ pairwise incomparable sets, but whenever we add to it a new set, we do find $k$ such sets. The size of the smallest such family is denoted by $\\text{sat}^*(n, \\mathcal A_{k})$. Ferrara, Kay, Kramer, Martin, Reiniger, Smith and Sullivan conjectured that $\\text{sat}^*(n, \\mathcal A_{k})=(k-1)n(1+o(1))$, and proved this for $k\\leq 4$. In this paper we prove this conjecture for $k=5$ and $k=6$. Moreover, we give the exact value for $\\text{sat}^*(n, \\mathcal A_5)$ and $\\text{sat}^*(n, \\mathcal A_6)$. We also give some open problems inspired by our analysis.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-15T22:39:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "06A07",
      "05D05"
    ],
    "keywords": [
      "small antichains",
      "saturation",
      "exact value",
      "open problems",
      "pairwise incomparable sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}