{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:four-uniform-case-of-morris-s-conjecture-mathrm-fc-4-n-theta-n-2",
  "title": "Four-uniform case of Morris’s conjecture: $\\mathrm{FC}(4,n)=\\Theta(n^2)$",
  "canonical_statement": "Morris conjectured that, for each fixed $k\\geq 2$, the Frankl-complete threshold satisfies $\\mathrm{FC}(k,n)=\\Theta(n^{k-2})$ as $n\\to\\infty$. Here $\\mathrm{FC}(k,n)$ is the least $m$ such that every collection of $m$ distinct $k$-subsets of an $n$-element set is Frankl-complete. A configuration $\\mathcal G$ with support $U=\\bigcup\\mathcal G$ is Frankl-complete if every finite union-closed family $\\mathcal F\\supseteq\\mathcal G$ has an element of $U$ in at least half its members. This entry concerns the four-uniform case: is $\\mathrm{FC}(4,n)=\\Theta(n^2)$?",
  "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:four-uniform-case-of-morris-s-conjecture-mathrm-fc-4-n-theta-n-2:event",
      "problem_id": "vibemathed:four-uniform-case-of-morris-s-conjecture-mathrm-fc-4-n-theta-n-2",
      "type": "proved",
      "status": "candidate",
      "occurred_at": "2026-09-01T00:00:00.000Z",
      "title": "Zenodo preprint (V1)",
      "summary": "It is proved that $\\mathrm{FC}(4,n)=\\Theta(n^2)$, settling the four-uniform case of Morris's conjecture; the three-uniform case had been settled exactly by Pulaj, so this was the first open uniformity. Explicitly, for $n\\ge 20$ and $r=\\lfloor n/6\\rfloor$, $r(n-3r)+1\\le\\mathrm{FC}(4,n)\\le 1+\\lfloor\\tfrac{160}{3}n(n-1)\\rfloor$, and with the Chung-Frankl triple-star theorem $\\mathrm{FC}(4,n)\\le(18+o(1))n^2$. The key step is a sharp sunflower criterion: four-sets with a common two-point core and pairwise disjoint two-point petals form a Frankl-complete configuration if and only if there are at least nine petals, the positive direction by a charging inequality with weight three on the core and one on the petals. Separately, exact certificates establish $\\mathrm{FC}(4,9)=16$ and refute Pulaj and Wood's lexicographic extremality conjecture. The paper also explains why higher-uniformity sunflowers cannot supply the same local forcing, so the general conjecture stays open.",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:four-uniform-case-of-morris-s-conjecture-mathrm-fc-4-n-theta-n-2:attempt"
      ],
      "method_family_ids": [
        "vibemathed:four-uniform-case-of-morris-s-conjecture-mathrm-fc-4-n-theta-n-2:method"
      ],
      "source_assertion_ids": [
        "vibemathed:four-uniform-case-of-morris-s-conjecture-mathrm-fc-4-n-theta-n-2:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:four-uniform-case-of-morris-s-conjecture-mathrm-fc-4-n-theta-n-2:attempt",
      "problem_id": "vibemathed:four-uniform-case-of-morris-s-conjecture-mathrm-fc-4-n-theta-n-2",
      "model": "GPT-6 Astra; Fable 5.1",
      "provider": "OpenAI; Anthropic",
      "attempted_at": "2026-09-01T00:00:00.000Z",
      "human_collaborators": [
        "Mingchang Liu"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The manuscript's AI statement, in full: \"The author and AI models both made substantial mathematical contributions to this work. The finite classification establishing $\\mathrm{FC}(4,9)=16$ was developed primarily by the models. The author proposed extending this work to the four-set case of Morris's conjecture and contributed to proof development and refinement. The collaboration developed the charging inequality, $3:1$ core-petal weights, and matching negative certificates for the sharp nine-petal sunflower threshold, and combined this local criterion with extremal bounds to prove $\\mathrm{FC}(4,n)=\\Theta(n^2)$. GPT-6 Astra (OpenAI) and Fable 5.1 (Anthropic) contributed to proof exploration, computation, and review. The author takes full responsibility for all mathematical results and the contents of this manuscript.\" Co-developed on that account: the models carried the finite classification and shared the main argument, the author set the target and takes responsibility.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "V1 code and exact certificates",
          "url": "https://github.com/michaeliu4/frankl-complete-four-sets/releases/tag/V1",
          "kind": "code",
          "license": null
        },
        {
          "label": "Morris’s original conjecture (Conjecture 2)",
          "url": "https://arxiv.org/abs/math/0702348v1",
          "kind": "problem-record",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "Zenodo preprint (V1)",
          "url": "https://zenodo.org/records/22702735",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Four-uniform case of Morris’s conjecture: $\\mathrm{FC}(4,n)=\\Theta(n^2)$",
          "url": "https://vibemathed.com/problem/four-uniform-case-of-morris-s-conjecture-mathrm-fc-4-n-theta-n-2",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:four-uniform-case-of-morris-s-conjecture-mathrm-fc-4-n-theta-n-2:verification",
      "problem_id": "vibemathed:four-uniform-case-of-morris-s-conjecture-mathrm-fc-4-n-theta-n-2",
      "solution_event_id": "vibemathed:four-uniform-case-of-morris-s-conjecture-mathrm-fc-4-n-theta-n-2:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Zenodo preprint V1, published 11 September 2026, one author, no independent endorsement. Checked here: the record exists with the abstract as submitted; the manuscript states Morris's Conjecture 2 as the entry does and records that Pulaj settled the three-uniform case exactly ($\\mathrm{FC}(3,n)=\\lfloor n/2\\rfloor+1$ for $n\\ge 4$), so this is the first open uniformity and not a duplicate of anything in the catalog; the companion repository release V1 holds the exact certificates for $\\mathrm{FC}(4,9)=16$ and for the Pulaj-Wood refutation with Python and C++ verifiers. None of the mathematics was checked here and the verifiers were not run. Filed as Candidate, as submitted, until an independent reader has been through the argument or the finite certificates have been replayed.",
      "sources": [
        {
          "label": "VibeMathed: Four-uniform case of Morris’s conjecture: $\\mathrm{FC}(4,n)=\\Theta(n^2)$",
          "url": "https://vibemathed.com/problem/four-uniform-case-of-morris-s-conjecture-mathrm-fc-4-n-theta-n-2",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "Zenodo preprint (V1)",
          "url": "https://zenodo.org/records/22702735",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "Zenodo preprint (V1)",
      "url": "https://zenodo.org/records/22702735",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Four-uniform case of Morris’s conjecture: $\\mathrm{FC}(4,n)=\\Theta(n^2)$",
      "url": "https://vibemathed.com/problem/four-uniform-case-of-morris-s-conjecture-mathrm-fc-4-n-theta-n-2",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:four-uniform-case-of-morris-s-conjecture-mathrm-fc-4-n-theta-n-2:signal:verify_now",
      "problem_id": "vibemathed:four-uniform-case-of-morris-s-conjecture-mathrm-fc-4-n-theta-n-2",
      "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"
}