{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-clique-and-independent-set",
  "semantic_family_id": "sf-clique-and-independent-set",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "Clique與獨立集"
  },
  "summary": {
    "zh-Hant": "Clique內每對不同頂點相鄰，獨立集內每對不同頂點不相鄰；最大與極大不可混同。"
  },
  "statement": {
    "zh-Hant": "候選集合逐對核相鄰或不相鄰。單一大小r候選只給omega或alpha的下界；宣稱maximum須遍歷全部頂點子集合或另有全域上界。maximal只代表不能加入頂點，且clique大小只推出chi(G)>=omega(G)。"
  },
  "formula": {
    "tex": "omega(G)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>omega</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "omega",
        "glyph": "omega"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "G",
          "glyph": "G"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "2ec10d06ca811919e8e398d931c48d311851b69bb32cf9824d075e7edaa8fe99"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "完整有限無loop簡單無向圖及無重複頂點子集。"
    },
    {
      "id": "a2",
      "zh-Hant": "exact extrema題限定n<=9並遍歷全部2^n子集。"
    }
  ],
  "symbols": [
    {
      "token": "omega",
      "role": "clique_number",
      "role_zh": "最大clique大小",
      "scope": "representative_formula"
    },
    {
      "token": "alpha",
      "role": "independence_number",
      "role_zh": "最大獨立集大小",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-graph",
      "title_zh": "有限簡單無向圖",
      "reason_zh": "相鄰關係。"
    },
    {
      "id": "mko-subset",
      "title_zh": "子集合",
      "reason_zh": "候選頂點集合。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "clique與independent set分別核全部不同頂點對。",
      "maximal是局部不可擴張，maximum是全域最大基數。",
      "clique給色數下界但不一般給等號。"
    ],
    "common_misconception_zh": "把任一極大clique當最大clique，或由omega直接宣稱chi=omega。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "全域extremum證書限小型完整圖；不宣稱大型圖高效演算法。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入Ramsey bounds與perfect graphs。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-clique-and-independent-set.md 整理；2026-10-14 脈衝 R02（使用者手動接續 2026-11-06 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}