{
  "id": "1803.02809",
  "version": "v1",
  "published": "2018-03-07T18:36:42.000Z",
  "updated": "2018-03-07T18:36:42.000Z",
  "title": "The size of the giant component in random hypergraphs: a short proof",
  "authors": [
    "Oliver Cooley",
    "Mihyun Kang",
    "Christoph Koch"
  ],
  "comment": "12 pages + 4 pages appendix",
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "We consider connected components in $k$-uniform hypergraphs for the following notion of connectedness: given integers $k\\ge 2$ and $1\\le j \\le k-1$, two $j$-sets (of vertices) lie in the same $j$-component if there is a sequence of edges from one to the other such that consecutive edges intersect in at least $j$ vertices. We prove that certain collections of $j$-sets constructed during a breadth-first search process on $j$-components in a random $k$-uniform hypergraph are reasonably regularly distributed with high probability. We use this property to provide a short proof of the asymptotic size of the giant $j$-component shortly after it appears.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-07T18:36:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C65",
      "05C80"
    ],
    "keywords": [
      "short proof",
      "giant component",
      "random hypergraphs",
      "uniform hypergraph",
      "breadth-first search process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}