{
  "id": "thresholds-and-expectation-thresholds",
  "title": "Thresholds and expectation thresholds",
  "description": "OpenAI's proofs (2026) of four conjectures on thresholds: the second Kahn–Kalai conjecture (containment thresholds of graphs within a logarithm of the expectation threshold), Talagrand's conjecture that integral and fractional expectation thresholds agree up to a constant, Talagrand's discrete convexity conjecture, and the Ascoli–He–Park–Talagrand decomposition of graphs into pieces that appear at a constant multiple of the integral expectation threshold. Planted resampling, tree covering, multiscale selectors, biased Fourier weights and embedding arguments, all proved in the course.",
  "units": [
    {
      "id": "random-sets-covers-and-expectation-thresholds",
      "title": "Random sets, covers and expectation thresholds",
      "reader": "random-sets-covers-and-expectation-thresholds.html",
      "source": "src/random-sets-covers-and-expectation-thresholds.md",
      "source_sha256": "23cd0dc052ced74a593f931690d019029e99e1067636fde0a3068500d9a77c38",
      "status": "author_self_checked"
    },
    {
      "id": "resampling-a-spread-family",
      "title": "Resampling a spread family",
      "reader": "resampling-a-spread-family.html",
      "source": "src/resampling-a-spread-family.md",
      "source_sha256": "d564fcc7722604b0dd724f01bc0e96cd8434bd9c9f37b630bfe5a64a29a12af8",
      "status": "author_self_checked"
    },
    {
      "id": "covering-a-probability-tree",
      "title": "Covering a probability tree",
      "reader": "covering-a-probability-tree.html",
      "source": "src/covering-a-probability-tree.md",
      "source_sha256": "590bbb5259ed3985fed848a8e3db9a4c37d7c54378ecd9cc9d8e7e1f323b3afb",
      "status": "author_self_checked"
    },
    {
      "id": "the-second-kahn-kalai-conjecture",
      "title": "The second Kahn–Kalai conjecture",
      "reader": "the-second-kahn-kalai-conjecture.html",
      "source": "src/the-second-kahn-kalai-conjecture.md",
      "source_sha256": "3029fd7dade696aff545a79852fb0729040f2670da73abaa025ad5ba6fd4f8d1",
      "status": "author_self_checked"
    },
    {
      "id": "integral-and-fractional-expectation-thresholds",
      "title": "Integral and fractional expectation thresholds",
      "reader": "integral-and-fractional-expectation-thresholds.html",
      "source": "src/integral-and-fractional-expectation-thresholds.md",
      "source_sha256": "d975c8d82d2eff0bb101c853f575ec77b86cfe581547aa78c6b2eb88d8818e8c",
      "status": "author_self_checked"
    },
    {
      "id": "the-discrete-convexity-conjecture",
      "title": "The discrete convexity conjecture",
      "reader": "the-discrete-convexity-conjecture.html",
      "source": "src/the-discrete-convexity-conjecture.md",
      "source_sha256": "c622de7500e04286ffcb43504f703a51fb36ec34ce24a2ac32939a9cd668bd0d",
      "status": "author_self_checked"
    },
    {
      "id": "embedding-sparse-graphs-in-random-graphs",
      "title": "Embedding sparse graphs in random graphs",
      "reader": "embedding-sparse-graphs-in-random-graphs.html",
      "source": "src/embedding-sparse-graphs-in-random-graphs.md",
      "source_sha256": "476c3b9966769d8845efa58b72d40705e3eba8d807cca9cd465e8c8076e5c052",
      "status": "author_self_checked"
    },
    {
      "id": "graph-decompositions-at-the-expectation-threshold",
      "title": "Graph decompositions at the expectation threshold",
      "reader": "graph-decompositions-at-the-expectation-threshold.html",
      "source": "src/graph-decompositions-at-the-expectation-threshold.md",
      "source_sha256": "bc98090f63221d08ecc588ff446fada242ef512ee87dc1654f7a98f703ce6882",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "the-gaussian-moat-problem lesson concentration-and-the-discrete-cube (Theorem 2.2, Hoeffding's inequality)",
    "amenability-and-unitarizability lesson contracting-averages-and-sparse-assignments (Lemma 2.1, Hall's theorem)",
    "sharp-thresholds-for-graph-properties lesson biased-fourier-analysis-and-hypercontractivity (Section 1, for comparison only)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}