{
  "id": "2104.03260",
  "version": "v1",
  "published": "2021-04-07T17:07:49.000Z",
  "updated": "2021-04-07T17:07:49.000Z",
  "title": "Intersecting families of sets are typically trivial",
  "authors": [
    "József Balogh",
    "Ramon I. Garcia",
    "Lina Li",
    "Adam Zsolt Wagner"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A family of subsets of $[n]$ is intersecting if every pair of its sets intersects. Determining the structure of large intersecting families is a central problem in extremal combinatorics. Frankl-Kupavskii and Balogh-Das-Liu-Sharifzadeh-Tran independently showed that for $n\\geq 2k + c\\sqrt{k\\ln k}$, almost all $k$-uniform intersecting families are stars. Improving their result, we show that the same conclusion holds for $n\\geq 2k+ 100\\ln k$. Our proof uses, among others, Sapozhenko's graph container lemma and the Das-Tran removal lemma.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-07T17:07:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "typically trivial",
      "sapozhenkos graph container lemma",
      "das-tran removal lemma",
      "large intersecting families",
      "central problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}