{
  "id": "seymours-second-neighborhood-conjecture",
  "title": "Seymour's second neighborhood conjecture",
  "description": "OpenAI's proof (2026) of Seymour's second neighborhood conjecture, that every finite oriented graph has a vertex with at least as many second as first out-neighbors; through minimal counterexamples, transitive tournament blow-ups, a pruning theorem proved with generic ranks, and an extremal argument; with weighted forms and consequences for directed triangles.",
  "units": [
    {
      "id": "second-out-neighborhoods-and-minimal-counterexamples",
      "title": "Second out-neighborhoods and minimal counterexamples",
      "reader": "second-out-neighborhoods-and-minimal-counterexamples.html",
      "source": "src/second-out-neighborhoods-and-minimal-counterexamples.md",
      "source_sha256": "7861b08abc74baffa96cea16b20b6b96062b8c46f7fb51c672544a806e4c3cf0",
      "status": "author_self_checked"
    },
    {
      "id": "blowing-up-by-transitive-tournaments",
      "title": "Blowing up by transitive tournaments",
      "reader": "blowing-up-by-transitive-tournaments.html",
      "source": "src/blowing-up-by-transitive-tournaments.md",
      "source_sha256": "af08a2cae8ff6be0e8eaac6f1a78824f55b90b8392245298bb7e8c85026bc47b",
      "status": "author_self_checked"
    },
    {
      "id": "ranks-matchings-and-pruning",
      "title": "Ranks, matchings and pruning",
      "reader": "ranks-matchings-and-pruning.html",
      "source": "src/ranks-matchings-and-pruning.md",
      "source_sha256": "7798955ea849998bca902bb58cd0e98ecd66eab5593c6b0f02bd0791d784899c",
      "status": "author_self_checked"
    },
    {
      "id": "the-extremal-argument-and-its-consequences",
      "title": "The extremal argument and its consequences",
      "reader": "the-extremal-argument-and-its-consequences.html",
      "source": "src/the-extremal-argument-and-its-consequences.md",
      "source_sha256": "02707392fe8e8de6be8abff59cf8115b53164e06085109e97a17e41c2b3823f5",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}