{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:erdos-matching-conjecture-the-four-uniform-case",
  "title": "Erdős matching conjecture: the four-uniform case",
  "canonical_statement": "Let $r\\geq2$, $s\\geq1$ and $n\\geq r(s+1)$ be integers. If $\\mathcal F\\subseteq\\binom{[n]}r$ contains no $s+1$ pairwise disjoint members, must\n\n$$\n|\\mathcal F|\\leq\\max\\left\\{\\binom{r(s+1)-1}{r},\\binom nr-\\binom{n-s}{r}\\right\\}?\n$$\n\nThe two candidate extremal families are all $r$-sets inside an $(r(s+1)-1)$-set and all $r$-sets meeting a fixed $s$-set. This is the Erdős matching conjecture (1965), recorded as Problem #1020. This entry concerns only the four-uniform specialization, $r=4$; it does not claim to settle arbitrary uniformity. The tracker's forbidden matching parameter is $k=s+1$.",
  "plain_summary": "VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "partial",
  "solution_events": [
    {
      "id": "vibemathed:erdos-matching-conjecture-the-four-uniform-case:event",
      "problem_id": "vibemathed:erdos-matching-conjecture-the-four-uniform-case",
      "type": "proved",
      "status": "partial",
      "occurred_at": "2026-09-01T00:00:00.000Z",
      "title": "GitHub manuscript: A computer-assisted proof of the four-uniform Erdős matching conjecture",
      "summary": "For $r=4$, the manuscript claims the sharp extremal value\n\n$$\n\\max\\left\\{\\binom{4s+3}{4},\\binom n4-\\binom{n-s}{4}\\right\\}\n$$\n\nfor every $s\\geq1$ and $n\\geq4s+4$. Both terms are attained by the standard clique and cover constructions. The claimed advance is an all-parameter four-uniform theorem, extending the sufficiently-large matching-number range of Hou–Hu–Liu ($s\\geq6961$) while retaining and crediting their framework and eight computations. The proof combines finite exact certificates and exhaustive searches with a written combinatorial reduction and analytic propagation.\n\nAdditional equality and stability conclusions have restricted domains; no classification of every extremizer or arbitrary-family stability is claimed. This is a claimed complete r=4 result and a partial result toward the general Erdős matching conjecture. Uniformity five and arbitrary uniformity are not settled by this paper.",
      "ai_contribution": "ai_assisted",
      "attempt_ids": [
        "vibemathed:erdos-matching-conjecture-the-four-uniform-case:attempt"
      ],
      "method_family_ids": [
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      "source_assertion_ids": [
        "vibemathed:erdos-matching-conjecture-the-four-uniform-case:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:erdos-matching-conjecture-the-four-uniform-case:attempt",
      "problem_id": "vibemathed:erdos-matching-conjecture-the-four-uniform-case",
      "model": "GPT-5.6 Sol; Astra; GPT-5 Pro; GPT-6 Pro",
      "provider": "OpenAI",
      "attempted_at": "2026-09-01T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The author reports using GPT-5.6 Sol, Astra, GPT-5 Pro and GPT-6 Pro through ChatGPT. Under the author's direction, AI systems explored proof approaches, proposed lemmas and counterchecks, developed exact certificates and verification code, drafted and revised the manuscript, and conducted adversarial reviews. The public manuscript explicitly discloses scientific and computational assistance, beyond language editing. The model names are supplied by the author; the manuscript does not identify their versions or assign individual results to particular models. AI-assisted is the conservative classification on this disclosure. The author selected the research scope and is responsible for the final claims. Reviews within this workflow are not independent expert verification.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Paper source, certificates and reproduction instructions",
          "url": "https://github.com/aconsciousfractal/Four-Uniform-Erdos-Matching-Conjecture/tree/60ce10893953bcf3ac0642d992b18dc0ac41d5b9",
          "kind": "code",
          "license": null
        },
        {
          "label": "Erdős Problem #1020: original problem and current scope",
          "url": "https://www.erdosproblems.com/1020",
          "kind": "problem-record",
          "license": null
        },
        {
          "label": "Hou–Hu–Liu: the four-uniform bound for s >= 6961",
          "url": "https://arxiv.org/abs/2605.26060v1",
          "kind": "paper",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "GitHub manuscript: A computer-assisted proof of the four-uniform Erdős matching conjecture",
          "url": "https://github.com/aconsciousfractal/Four-Uniform-Erdos-Matching-Conjecture/blob/60ce10893953bcf3ac0642d992b18dc0ac41d5b9/paper/Four-Uniform-Erdos-Matching-Conjecture.pdf",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Erdős matching conjecture: the four-uniform case",
          "url": "https://vibemathed.com/problem/erdos-matching-conjecture-the-four-uniform-case",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:erdos-matching-conjecture-the-four-uniform-case:verification",
      "problem_id": "vibemathed:erdos-matching-conjecture-the-four-uniform-case",
      "solution_event_id": "vibemathed:erdos-matching-conjecture-the-four-uniform-case:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Checked here on 13 September 2026, at the sources rather than from the submission. erdosproblems.com/1020 carries the conjecture in the form stated and still marks it open; the tracker's forbidden-matching parameter is $k$ where this entry uses $s+1$, and substituting turns one formula into the other, so the transcription is faithful. Hou, Hu and Liu (arXiv:2605.26060) state in their abstract \"We prove the 4-uniform Erdos Matching Conjecture for every matching number $s\\ge 6961$\", so the standing state of the four-uniform case really was all sufficiently large $s$, and the claimed advance - closing the remaining finite range - is well defined.\n\nThe mathematics was not checked. The author's own audit of 12 September records replay of all 27 computational obligations including the eight retained Hou-Hu-Liu searches, 52 tests, 12 receipt-integrity mutation controls, a matching PDF rebuild and an anonymous clone matching all 97 reviewed files. That supports reproducibility and the declared finite computations; it does not validate the written reduction and induction, and no independent domain expert endorsement or formal proof is supplied. Announcement reflects repository-only publication.",
      "sources": [
        {
          "label": "VibeMathed: Erdős matching conjecture: the four-uniform case",
          "url": "https://vibemathed.com/problem/erdos-matching-conjecture-the-four-uniform-case",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "GitHub manuscript: A computer-assisted proof of the four-uniform Erdős matching conjecture",
          "url": "https://github.com/aconsciousfractal/Four-Uniform-Erdos-Matching-Conjecture/blob/60ce10893953bcf3ac0642d992b18dc0ac41d5b9/paper/Four-Uniform-Erdos-Matching-Conjecture.pdf",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "GitHub manuscript: A computer-assisted proof of the four-uniform Erdős matching conjecture",
      "url": "https://github.com/aconsciousfractal/Four-Uniform-Erdos-Matching-Conjecture/blob/60ce10893953bcf3ac0642d992b18dc0ac41d5b9/paper/Four-Uniform-Erdos-Matching-Conjecture.pdf",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Erdős matching conjecture: the four-uniform case",
      "url": "https://vibemathed.com/problem/erdos-matching-conjecture-the-four-uniform-case",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "expand",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:erdos-matching-conjecture-the-four-uniform-case:signal:recent_partial_result",
      "problem_id": "vibemathed:erdos-matching-conjecture-the-four-uniform-case",
      "kind": "recent_partial_result",
      "reason": "A source-reported partial result may support bounded expansion.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "uncertainties": [
    "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"
}