{
  "id": "euclidean-ramsey-configurations",
  "title": "The classification of finite Euclidean Ramsey sets",
  "description": "OpenAI's classification (2026) of the finite Euclidean Ramsey sets: a finite set is Ramsey exactly when a linear system over the tensor square of its coordinate field has a solution. The colouring obstruction from rational functionals, a finite lattice identity through the flatness of completion, two-scale configurations by Gaussian smoothing, the Hales–Jewett theorem, ultrafilter semigroups and a path criterion in a free group, and the correction of endpoints through derived series. Examples on circles: at most five concyclic points are Ramsey, including kites that embed in no finite transitive set; nine generic points and three rotated squares are not.",
  "units": [
    {
      "id": "ramsey-sets-and-the-tensor-criterion",
      "title": "Ramsey sets and the tensor criterion",
      "reader": "ramsey-sets-and-the-tensor-criterion.html",
      "source": "src/ramsey-sets-and-the-tensor-criterion.md",
      "source_sha256": "e13baad970b8b0f54137781a5fdbbbb9357c50aadd9ffc8b357c04024206a883",
      "status": "author_self_checked"
    },
    {
      "id": "a-colouring-obstruction",
      "title": "A colouring obstruction",
      "reader": "a-colouring-obstruction.html",
      "source": "src/a-colouring-obstruction.md",
      "source_sha256": "64a8517c2ac8a4c28c1d4fadcd5359ee665237c7068f1f187c576b06f77f357e",
      "status": "author_self_checked"
    },
    {
      "id": "a-finite-lattice-identity",
      "title": "A finite lattice identity",
      "reader": "a-finite-lattice-identity.html",
      "source": "src/a-finite-lattice-identity.md",
      "source_sha256": "64fe60e2e7d7f0b47ae123b8b235afe9aacc65844d9b9f59fe2be7f7a6722ae3",
      "status": "author_self_checked"
    },
    {
      "id": "two-scale-configurations",
      "title": "Two-scale configurations",
      "reader": "two-scale-configurations.html",
      "source": "src/two-scale-configurations.md",
      "source_sha256": "7774726066d4d644da1a383687c42104d678d972bf180b22b92436b25582d422",
      "status": "author_self_checked"
    },
    {
      "id": "the-hales-jewett-theorem",
      "title": "The Hales–Jewett theorem",
      "reader": "the-hales-jewett-theorem.html",
      "source": "src/the-hales-jewett-theorem.md",
      "source_sha256": "9b52c728cdbad6f9a9b689d1dc87e12a21f000b15ba1355506d5213362e0768e",
      "status": "author_self_checked"
    },
    {
      "id": "ultrafilters-and-idempotents",
      "title": "Ultrafilters and idempotents",
      "reader": "ultrafilters-and-idempotents.html",
      "source": "src/ultrafilters-and-idempotents.md",
      "source_sha256": "85a70a8311650d747cbbe881298d7bb5c2b87b221c9e61c7bef9d97861a63c15",
      "status": "author_self_checked"
    },
    {
      "id": "the-path-criterion",
      "title": "The path criterion",
      "reader": "the-path-criterion.html",
      "source": "src/the-path-criterion.md",
      "source_sha256": "8d8da1dab2c1806eefb5dbf60b8addfe449c9f7b5d4ce9fa535033809b65e434",
      "status": "author_self_checked"
    },
    {
      "id": "equal-endpoints",
      "title": "Equal endpoints",
      "reader": "equal-endpoints.html",
      "source": "src/equal-endpoints.md",
      "source_sha256": "6b825ef3a96c696ea6ac2451b4b85d3601fa0f6bd390be24f1b3d501de1883d1",
      "status": "author_self_checked"
    },
    {
      "id": "examples-on-circles",
      "title": "Examples on circles",
      "reader": "examples-on-circles.html",
      "source": "src/examples-on-circles.md",
      "source_sha256": "ed094ee5aa1b91401c2d64cb1bbf37caddce58388d977fd92d173a1489583c43",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "linear algebra over a field, including Gaussian elimination and the diagonalization of real symmetric matrices; Zorn's lemma",
    "the programme's course Commutative algebra for geometry (AG-CA): the Hilbert basis theorem (lesson Noetherian and Artinian rings, Theorem 2.1) and the flatness of completion (lesson Completion, Theorem 3.2), used in the third lesson",
    "core courses Real Analysis I and II (Lebl, Basic Analysis I and II, version 6.3): Taylor's theorem (Theorem 4.3.2), the Leibniz integral rule (Theorem 9.1.1), Riemann integrability of continuous functions on rectangles (Theorem 10.1.15), Fubini's theorem (Theorem 10.2.2) and the change of variables (Theorem 10.7.2), used in the fourth lesson"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}