{
  "id": "1506.04686",
  "version": "v1",
  "published": "2015-06-15T18:03:53.000Z",
  "updated": "2015-06-15T18:03:53.000Z",
  "title": "Extremal Bounds for Bootstrap Percolation in the Hypercube",
  "authors": [
    "Natasha Morrison",
    "Jonathan A. Noel"
  ],
  "comment": "22 pages, 2 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "The $r$-neighbour bootstrap process on a graph $G$ starts with an initial set $A_0$ of \"infected\" vertices and, at each step of the process, a healthy vertex becomes infected if it has at least $r$ infected neighbours (once a vertex becomes infected, it remains infected forever). If every vertex of $G$ eventually becomes infected, then we say that $A_0$ percolates. We prove a conjecture of Balogh and Bollob\\'{a}s which says that, for fixed $r$ and $d\\to\\infty$, every percolating set in the $d$-dimensional hypercube has cardinality at least $\\frac{1+o(1)}{r}\\binom{d}{r-1}$. We also prove an analogous result for multidimensional rectangular grids. Our proofs exploit a connection between bootstrap percolation and a related process, known as weak saturation. In addition, we improve the best known upper bound for the minimum size of a percolating set in the hypercube. In particular, when $r=3$, we prove that the minimum cardinality of a percolating set in the $d$-dimensional hypercube is $\\left\\lceil\\frac{d(d+3)}{6}\\right\\rceil+1$ for all $d\\geq3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-15T18:03:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D99",
      "05C35",
      "82B43"
    ],
    "keywords": [
      "bootstrap percolation",
      "extremal bounds",
      "percolating set",
      "dimensional hypercube",
      "multidimensional rectangular grids"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150604686M"
    }
  }
}