{
  "id": "1908.01108",
  "version": "v1",
  "published": "2019-08-03T02:27:31.000Z",
  "updated": "2019-08-03T02:27:31.000Z",
  "title": "Improved bounds for induced poset saturation",
  "authors": [
    "Ryan R. Martin",
    "Heather C. Smith",
    "Shanise Walker"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Given a finite poset $\\mathcal{P}$, a family $\\mathcal{F}$ of elements in the Boolean lattice is induced-$\\mathcal{P}$-saturated if $\\mathcal{F}$ contains no copy of $\\mathcal{P}$ as an induced subposet but every proper superset of $\\mathcal{F}$ contains a copy of $\\mathcal{P}$ as an induced subposet. The minimum size of an induced-$\\mathcal{P}$-saturated family in the $n$-dimensional Boolean lattice, denoted $\\operatorname{sat}^*(n,\\mathcal{P})$, was first studied by Ferrara et al. (2017). Our work focuses on strengthening lower bounds. For the 4-point poset known as the diamond, we prove $\\operatorname{sat}^*(n,\\mathcal{D}_2)\\geq\\sqrt{n}$, improving upon a logarithmic lower bound. For the antichain with $k+1$ elements, we prove $\\operatorname{sat}^*(n,\\mathcal{A}_{k+1})\\geq (1-o_k(1))\\frac{kn}{\\log_2 k}$, improving upon a lower bound of $3n-1$ for $k\\geq 3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-03T02:27:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "06A07",
      "05D05"
    ],
    "keywords": [
      "induced poset saturation",
      "induced subposet",
      "dimensional boolean lattice",
      "logarithmic lower bound",
      "proper superset"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}