{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:big-line-big-clique-four-collinear-or-six-clique",
  "title": "The Kára–Pór–Wood Big-Line-Big-Clique Conjecture: Four Collinear Points or a Six-Clique",
  "canonical_statement": "The big-line-big-clique conjecture of Kára, Pór and Wood asserts that for all $k, \\ell$ there is an $n$ such that every finite point set of at least $n$ points contains $\\ell$ collinear points or $k$ points that pairwise see each other. True for $\\ell = 4$, $k = 6$, the first case left open: every finite point set of size at least $10^{11055931}$ has four collinear points or six pairwise visible points.",
  "plain_summary": "VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "candidate",
  "solution_events": [
    {
      "id": "vibemathed:big-line-big-clique-four-collinear-or-six-clique:event",
      "problem_id": "vibemathed:big-line-big-clique-four-collinear-or-six-clique",
      "type": "proved",
      "status": "candidate",
      "occurred_at": "2026-08-19T00:00:00.000Z",
      "title": "Large Finite Point Sets Have 4 Collinear Points or a 6-Clique",
      "summary": "VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:big-line-big-clique-four-collinear-or-six-clique:attempt"
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      "source_assertion_ids": [
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    {
      "id": "vibemathed:big-line-big-clique-four-collinear-or-six-clique:attempt",
      "problem_id": "vibemathed:big-line-big-clique-four-collinear-or-six-clique",
      "model": "GPT-5.6 Sol Pro",
      "provider": "OpenAI",
      "attempted_at": "2026-08-19T00:00:00.000Z",
      "human_collaborators": [
        "Édouard Bonnet"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The paper's disclosure, in full: \"After one fruitless attempt and the now customary generic encouragement, a relatively detailed proof of Theorem 1 was provided by GPT-5.6 Sol Pro after pondering for 222 minutes. The author's contributions were limited to checking the proof, simplifying some parts, and eventually writing the paper in a way that would give the author (and hopefully other human readers) a more pleasant reading experience.\"",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "Large Finite Point Sets Have 4 Collinear Points or a 6-Clique",
          "url": "https://arxiv.org/abs/2608.19468",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: The Kára–Pór–Wood Big-Line-Big-Clique Conjecture: Four Collinear Points or a Six-Clique",
          "url": "https://vibemathed.com/problem/big-line-big-clique-four-collinear-or-six-clique",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
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  "verifications": [
    {
      "id": "vibemathed:big-line-big-clique-four-collinear-or-six-clique:verification",
      "problem_id": "vibemathed:big-line-big-clique-four-collinear-or-six-clique",
      "solution_event_id": "vibemathed:big-line-big-clique-four-collinear-or-six-clique:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Checked by this site on 21 August 2026 against the paper (arXiv:2608.19468v1, 7pp): the disclosure is verbatim as quoted, the conjecture's 2005 attribution is in the text, and the paper states this is the first open case resolved since 2009. The proof was not checked here. The threshold is astronomical but the statement is unconditional. Days-old preprint, no independent review.",
      "sources": [
        {
          "label": "VibeMathed: The Kára–Pór–Wood Big-Line-Big-Clique Conjecture: Four Collinear Points or a Six-Clique",
          "url": "https://vibemathed.com/problem/big-line-big-clique-four-collinear-or-six-clique",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "Large Finite Point Sets Have 4 Collinear Points or a 6-Clique",
          "url": "https://arxiv.org/abs/2608.19468",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
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  ],
  "sources": [
    {
      "label": "Large Finite Point Sets Have 4 Collinear Points or a 6-Clique",
      "url": "https://arxiv.org/abs/2608.19468",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: The Kára–Pór–Wood Big-Line-Big-Clique Conjecture: Four Collinear Points or a Six-Clique",
      "url": "https://vibemathed.com/problem/big-line-big-clique-four-collinear-or-six-clique",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:big-line-big-clique-four-collinear-or-six-clique:signal:verify_now",
      "problem_id": "vibemathed:big-line-big-clique-four-collinear-or-six-clique",
      "kind": "verify_now",
      "reason": "The source labels this result as a candidate; verification should precede reuse.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "uncertainties": [
    "The source status is candidate and must not be represented as solved.",
    "VibeMath has not independently verified the mathematical claim.",
    "AI-attempt independence and training-data exposure are unknown unless explicitly documented.",
    "VibeMath has not independently audited the mathematical statement, proof, or novelty claim."
  ],
  "generated_at": "2026-09-13T16:28:40.178Z"
}