{
  "id": "2105.02985",
  "version": "v1",
  "published": "2021-05-06T21:55:08.000Z",
  "updated": "2021-05-06T21:55:08.000Z",
  "title": "Sharp threshold for the Erdős-Ko-Rado theorem",
  "authors": [
    "József Balogh",
    "Robert A. Krueger",
    "Haoran Luo"
  ],
  "comment": "27 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "For positive integers $n$ and $k$ with $n\\geq 2k+1$, the Kneser graph $K(n,k)$ is the graph with vertex set consisting of all $k$-sets of $\\{1,\\dots,n\\}$, where two $k$-sets are adjacent exactly when they are disjoint. Let $K_p(n,k)$ be a random spanning subgraph of $K(n,k)$ where each edge is included independently with probability $p$. Bollob\\'as, Narayanan, and Raigorodskii asked for what $p$ does $K_p(n,k)$ have the same independence number as $K(n,k)$ with high probability. For $n=2k+1$, we prove a hitting time result, which gives a sharp threshold for this problem at $p=3/4$. Additionally, completing work of Das and Tran and work of Devlin and Kahn, we determine a sharp threshold function for all $n>2k+1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-06T21:55:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80",
      "05D05",
      "05D40"
    ],
    "keywords": [
      "erdős-ko-rado theorem",
      "sharp threshold function",
      "kneser graph",
      "independence number",
      "high probability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}