{
  "id": "sharp-thresholds-for-graph-properties",
  "title": "Sharp thresholds for graph properties",
  "description": "OpenAI's proofs (2026) of the Friedgut–Kalai conjectures: increasing graph properties of G(n,p) have threshold width O(log(1/ε)/(log n)²), and r-uniform hypergraph properties width O((log n)^(−r/(r−1))). The course develops biased Fourier analysis and hypercontractivity, restricts symmetric functions to the edges meeting a random block of vertices, and captures each Fourier support at one vertex.",
  "units": [
    {
      "id": "biased-fourier-analysis-and-hypercontractivity",
      "title": "Biased Fourier analysis and hypercontractivity",
      "reader": "biased-fourier-analysis-and-hypercontractivity.html",
      "source": "src/biased-fourier-analysis-and-hypercontractivity.md",
      "source_sha256": "02c737c1e8a436aaeacae594abb6c6c3f2fccb58228c4aa82f61d3562ee17f2e",
      "status": "author_self_checked"
    },
    {
      "id": "vertex-blocks-and-graph-properties",
      "title": "Vertex blocks and graph properties",
      "reader": "vertex-blocks-and-graph-properties.html",
      "source": "src/vertex-blocks-and-graph-properties.md",
      "source_sha256": "ec6db2e81fccb35d741ebb8c4814b0232ed7e5db4773145ca96cd9bdbe5a65a4",
      "status": "author_self_checked"
    },
    {
      "id": "hypergraph-properties",
      "title": "Hypergraph properties",
      "reader": "hypergraph-properties.html",
      "source": "src/hypergraph-properties.md",
      "source_sha256": "e26194aebb170058004d0ab73a34ecadd14417a14b4fb2a793633b6bffaf2075",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}