{
  "id": "crossing-numbers-of-complete-graphs",
  "title": "Crossing numbers of complete and complete bipartite graphs",
  "description": "Zarankiewicz's formula for cr(K_{m,n}) and the Harary-Hill formula for cr(K_n), proved for unrestricted drawings: normal drawings, the cycle pairing of signed crossings, interpolation, a subspace intersection inequality, and the matching drawings (OpenAI, 2026).",
  "units": [
    {
      "id": "drawings-and-crossing-numbers",
      "title": "Drawings and crossing numbers",
      "reader": "drawings-and-crossing-numbers.html",
      "source": "src/drawings-and-crossing-numbers.md",
      "source_sha256": "2b395d180317934bf6c3cc3541bff766da3b1b7d96a9d675ea9217ff87d29c7f",
      "status": "author_self_checked"
    },
    {
      "id": "polygonal-drawings",
      "title": "Polygonal drawings",
      "reader": "polygonal-drawings.html",
      "source": "src/polygonal-drawings.md",
      "source_sha256": "0e74d9ca0c7a0e0798b5b01dcd035b1c8c6699809fffaa7afff522fdd0e8ed52",
      "status": "author_self_checked"
    },
    {
      "id": "signed-crossings-and-cycles",
      "title": "Signed crossings and cycles",
      "reader": "signed-crossings-and-cycles.html",
      "source": "src/signed-crossings-and-cycles.md",
      "source_sha256": "280bd6f57eb6cd1b16fef4be1b60f686c2e6d316cb43716fc6f76911af790762",
      "status": "author_self_checked"
    },
    {
      "id": "interpolation",
      "title": "Interpolation",
      "reader": "interpolation.html",
      "source": "src/interpolation.md",
      "source_sha256": "5958e2d7eed20954548e34ae3fd06210ed74465b334c912d621f433fe774a8bb",
      "status": "author_self_checked"
    },
    {
      "id": "an-intersection-inequality-for-subspaces",
      "title": "An intersection inequality for subspaces",
      "reader": "an-intersection-inequality-for-subspaces.html",
      "source": "src/an-intersection-inequality-for-subspaces.md",
      "source_sha256": "bab61337426725766716f9797b07b4523c5268c75f587f19af59b0065e3fcb90",
      "status": "author_self_checked"
    },
    {
      "id": "the-zarankiewicz-formula",
      "title": "The Zarankiewicz formula",
      "reader": "the-zarankiewicz-formula.html",
      "source": "src/the-zarankiewicz-formula.md",
      "source_sha256": "5b0a78a31d13de0f4021b3709d10622c523abf6c5c786e439686bbe8eae4d131",
      "status": "author_self_checked"
    },
    {
      "id": "the-lower-bound-for-complete-graphs",
      "title": "The lower bound for complete graphs",
      "reader": "the-lower-bound-for-complete-graphs.html",
      "source": "src/the-lower-bound-for-complete-graphs.md",
      "source_sha256": "46fe20612fa5720f77a9582620013de84ed3feb80f6695e22828b69b6f61a34b",
      "status": "author_self_checked"
    },
    {
      "id": "two-page-drawings-and-the-harary-hill-formula",
      "title": "Two-page drawings and the Harary–Hill formula",
      "reader": "two-page-drawings-and-the-harary-hill-formula.html",
      "source": "src/two-page-drawings-and-the-harary-hill-formula.md",
      "source_sha256": "bae7c9ecf9282d4d93c8822c3e73079c8c54e344c6da0891c90491d50432c601",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Real Analysis I and II (Lebl, Basic Analysis I & II, version 6.3): Lemma 7.4.10 (Lebesgue covering lemma), Theorem 7.4.14 (Heine-Borel), Lemma 7.5.5, Theorem 7.5.6 and Theorem 8.5.1 (inverse function theorem)",
    "core course Linear Algebra (Hefferon, Linear Algebra, fourth edition): Theorem Two.III.3.11, Theorem Three.II.2.14, Lemma Four.I.2.4, Definition Four.I.3.12, Corollary Four.I.3.16 and Theorem Four.II.1.5",
    "core course Abstract Algebra I (Judson, Abstract Algebra: Theory and Applications): Proposition 17.1.4 and Corollaries 17.2.3 and 17.2.4"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}