{
  "id": "one-sample-matroid-prophet-inequalities",
  "title": "One sample suffices for matroid prophet inequalities",
  "description": "OpenAI's theorem (2026) that one independent sample per element suffices for a constant-competitive prophet inequality on every finite matroid, even when the arrival order is chosen by an adversary who sees all samples, values and the rule's random seed; through density expansions, guarded paths of flats, reverse exposure and a marked-pivot count.",
  "units": [
    {
      "id": "prophets-matroids-and-an-almighty-adversary",
      "title": "Prophets, matroids and an almighty adversary",
      "reader": "prophets-matroids-and-an-almighty-adversary.html",
      "source": "src/prophets-matroids-and-an-almighty-adversary.md",
      "source_sha256": "627000d4eff2dc25e04fb910d0769ff934480ac734dc684a672657afeb2a02ff",
      "status": "author_self_checked"
    },
    {
      "id": "rounding-a-maximum-rule-and-candidates",
      "title": "Rounding, a maximum rule and candidates",
      "reader": "rounding-a-maximum-rule-and-candidates.html",
      "source": "src/rounding-a-maximum-rule-and-candidates.md",
      "source_sha256": "7eb20ba9d04aa29daa31dfc9d36f239dbb6495d930584fc6b7e6bdc3894ea429",
      "status": "author_self_checked"
    },
    {
      "id": "density-expansions-and-paths-of-flats",
      "title": "Density expansions and paths of flats",
      "reader": "density-expansions-and-paths-of-flats.html",
      "source": "src/density-expansions-and-paths-of-flats.md",
      "source_sha256": "3a1c1a24d7106c8e95d58633b185457e1e6bdd76e1aa2411e219c8a6bc2262b2",
      "status": "author_self_checked"
    },
    {
      "id": "layers-and-a-rank-bound-for-every-order",
      "title": "Layers and a rank bound for every order",
      "reader": "layers-and-a-rank-bound-for-every-order.html",
      "source": "src/layers-and-a-rank-bound-for-every-order.md",
      "source_sha256": "bc8e03bafae0a0573c2005f8b6dfc0c29d3f1cfa6d12927a9b5c063902821da8",
      "status": "author_self_checked"
    },
    {
      "id": "disjoint-certificates",
      "title": "Disjoint certificates",
      "reader": "disjoint-certificates.html",
      "source": "src/disjoint-certificates.md",
      "source_sha256": "cf20714ecc58f6f50a9420fe402917fc2b0c9f4d83e38355978cc2452c7657ac",
      "status": "author_self_checked"
    },
    {
      "id": "reverse-exposure-and-safe-samples",
      "title": "Reverse exposure and safe samples",
      "reader": "reverse-exposure-and-safe-samples.html",
      "source": "src/reverse-exposure-and-safe-samples.md",
      "source_sha256": "d429434ca39c0d908094470c9bf7f8b05c072669e3e00031ffac06f2dbd0cd7e",
      "status": "author_self_checked"
    },
    {
      "id": "marked-pivots-and-sign-patterns",
      "title": "Marked pivots and sign patterns",
      "reader": "marked-pivots-and-sign-patterns.html",
      "source": "src/marked-pivots-and-sign-patterns.md",
      "source_sha256": "81422999125e9f070f29356d3bad0dbfa640e23001379fe2c709f4a4c93fa30f",
      "status": "author_self_checked"
    },
    {
      "id": "from-safe-samples-to-unrevealed-rank",
      "title": "From safe samples to unrevealed rank",
      "reader": "from-safe-samples-to-unrevealed-rank.html",
      "source": "src/from-safe-samples-to-unrevealed-rank.md",
      "source_sha256": "c27914c0a31378bc8a27bb3e2aadfae324f490d529b104a34b0e28346f841573",
      "status": "author_self_checked"
    },
    {
      "id": "the-final-count-and-a-secretary-rule",
      "title": "The final count and a secretary rule",
      "reader": "the-final-count-and-a-secretary-rule.html",
      "source": "src/the-final-count-and-a-secretary-rule.md",
      "source_sha256": "fd56c2fb94db4b207406195971eaffc54287611caaa74e68fe2a01c79386036d",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "finite sets, independence of finitely many random variables and conditional expectation over finite probability spaces; random values with arbitrary laws on [0, infinity) enter only through expectations of nonnegative measurable functions",
    "the course proves all matroid facts it uses"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}