{
  "id": "KL-FCS-046",
  "version": "1.0.0",
  "domain": "Constraint satisfaction",
  "kind": "constraints",
  "title": "A triangle cannot use two colors",
  "problem": "Decide and count color assignments satisfying every graph edge.",
  "specification": {
    "vertices": 3,
    "edges": [
      [
        0,
        1
      ],
      [
        1,
        2
      ],
      [
        2,
        0
      ]
    ],
    "colors": 2
  },
  "claim": {
    "satisfiable": false,
    "solution_count": 0
  },
  "witness": {
    "method": "exhaustive enumeration"
  },
  "verification_scope": "Complete assignment space · 8 candidates",
  "explanation": "Every assignment either yields a fully valid coloring or an edge witnessing failure. Counting includes distinct color labels, so symmetric assignments remain separate.",
  "limitations": "This is a finite coloring instance; the solution count is not reduced by graph or color symmetries.",
  "verification_status": "mechanically-checked",
  "review_status": "awaiting-independent-review",
  "provenance": {
    "origin": "Original Kenton Labs reference instance, authored with Codex assistance on 2026-10-11.",
    "external_dataset": null,
    "model_run": null
  },
  "references": [
    "https://ocw.mit.edu/courses/18-404j-theory-of-computation-fall-2020/download/"
  ],
  "license_status": "not-yet-selected",
  "dataset": {
    "family": "constraints",
    "task": "Decide and count color assignments satisfying every graph edge.",
    "input_encoding": "Structured JSON; field meanings are stated in the specification.",
    "coverage": "Complete assignment space · 8 candidates",
    "acceptance": [
      "Enumerate one color value for each vertex.",
      "Check every edge’s unequal-color constraint.",
      "Count the complete solution space.",
      "Validate a coloring witness, or require enumeration evidence when no model exists."
    ],
    "generation": "Deterministic finite fixture; full enumeration or witness replay as stated.",
    "split_policy": "Reference corpus for exposition and reproduction; no train/test evaluation split is claimed."
  },
  "lesson": {
    "motivation": "Expose both satisfiable assignments and complete impossibility arguments for finite variable domains.",
    "definitions": [
      {
        "term": "Constraint",
        "definition": "A relation restricting allowed variable assignments."
      },
      {
        "term": "Graph coloring",
        "definition": "Adjacent vertices must receive different colors."
      },
      {
        "term": "Solution space",
        "definition": "Every assignment satisfying all declared constraints."
      }
    ],
    "reasoning": [
      "Enumerate one color value for each vertex.",
      "Check every edge’s unequal-color constraint.",
      "Count the complete solution space.",
      "Validate a coloring witness, or require enumeration evidence when no model exists."
    ],
    "worked_example": "Every assignment either yields a fully valid coloring or an edge witnessing failure. Counting includes distinct color labels, so symmetric assignments remain separate.",
    "complexity": "k colors on n vertices produce kⁿ assignments. Color-label permutations can create symmetric solutions.",
    "common_error": "Failing to find a solution is not equivalent to proving there is none.",
    "further_work": "Add symmetry reduction and checked propagation or conflict explanations."
  },
  "related_ids": [
    "KL-FCS-047",
    "KL-FCS-048"
  ]
}