{
  "id": "large-independent-sets-in-three-colorable-graphs",
  "title": "Large independent sets in three-colorable graphs",
  "description": "Label Cover instances; a graph built from families of phases on layers of tuples of questions, with a proper coloring by three colors when all tests can be satisfied. Lipschitz functions for the sup distance on tori: odd extension, means over coordinates, influences. Lists of answers from an independent set, and the alignment theorem for families of lists, with proofs of the minimax theorem and of Ramsey's theorem. The theorem: for every delta > 0 it is NP-hard to tell graphs with a proper coloring by three colors from graphs whose independent sets have fewer than delta |V| vertices, from two named hypotheses (the PCP theorem and parallel repetition); Austin's theorem on Lipschitz functions of many variables, which the proof uses, is proved in the course Hypercontractivity on a torus and juntas for Lipschitz functions. The course is an exposition of OpenAI's proof (2026).",
  "units": [
    {
      "id": "label-cover-and-the-graph-of-the-reduction",
      "title": "Label Cover and the graph of the reduction",
      "reader": "label-cover-and-the-graph-of-the-reduction.html",
      "source": "src/label-cover-and-the-graph-of-the-reduction.md",
      "source_sha256": "18d42257b8321e201c7ffff9f060c429354d900112e0ba9cd091c55be662353c",
      "status": "author_self_checked"
    },
    {
      "id": "lipschitz-functions-on-a-torus-extension-juntas-and-influences",
      "title": "Lipschitz functions on a torus: extension, juntas and influences",
      "reader": "lipschitz-functions-on-a-torus-extension-juntas-and-influences.html",
      "source": "src/lipschitz-functions-on-a-torus-extension-juntas-and-influences.md",
      "source_sha256": "879a61ab589e486c561c2420ba7f317af40e2af4065d4aca20c3f4b1e0b685e4",
      "status": "author_self_checked"
    },
    {
      "id": "from-an-independent-set-to-lists-of-answers",
      "title": "From an independent set to lists of answers",
      "reader": "from-an-independent-set-to-lists-of-answers.html",
      "source": "src/from-an-independent-set-to-lists-of-answers.md",
      "source_sha256": "0a9df8f902e418da59f7ccfbfb87c7eab5ac1deacab4373a978b66fe5fdcdfbf",
      "status": "author_self_checked"
    },
    {
      "id": "alignment-of-lists-patterns-minimax-and-ramseys-theorem",
      "title": "Alignment of lists: patterns, minimax and Ramsey's theorem",
      "reader": "alignment-of-lists-patterns-minimax-and-ramseys-theorem.html",
      "source": "src/alignment-of-lists-patterns-minimax-and-ramseys-theorem.md",
      "source_sha256": "a49709d155ff533cc92822ad7735f2e7521a57e612e59a2fa43e83ef5d23048f",
      "status": "author_self_checked"
    },
    {
      "id": "order-invariant-arrays-and-the-common-index",
      "title": "Order-invariant arrays and the common index",
      "reader": "order-invariant-arrays-and-the-common-index.html",
      "source": "src/order-invariant-arrays-and-the-common-index.md",
      "source_sha256": "9008097b68e19c39563a0a1735315f236f7aee7e4cba958cd8afd7c3500c39c6",
      "status": "author_self_checked"
    },
    {
      "id": "the-coefficient-test-and-the-theorem",
      "title": "The coefficient test and the theorem",
      "reader": "the-coefficient-test-and-the-theorem.html",
      "source": "src/the-coefficient-test-and-the-theorem.md",
      "source_sha256": "83dbbd4f5e1fd0c8f867d1a9662b5587664c04d758b24f3c45d1e08672ce0c29",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "the lesson 'Projection games and the base game' of the course 'Two-to-one games with perfect completeness' (Theorem 5.1 there, from the named hypotheses (PCP) and (PR))",
    "core course Measure and Integration (D10): the Lebesgue integral on a cube, Fubini's theorem, countable additivity of the Lebesgue measure on [0,1)",
    "finite probability; compact sets in R^n",
    "the lesson 'Lipschitz functions are close to juntas' of the course 'Hypercontractivity on a torus and juntas for Lipschitz functions' (Theorem 4.1 there is Theorem (J) of the second lesson)",
    "the lessons prove every other fact they use"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}