{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1736",
  "content_hash": null,
  "slug": "ryser-hypergraph-cover-conjecture-attempt-dated-source-audit",
  "type": "attempt",
  "title": "Dated source and duplicate audit",
  "summary": "The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: The conjecture is proved for all hypergraphs when r≤3 and for intersecting hypergraphs when r≤5. For r=4 and r=5, Haxell and Scott prove τ(H)≤(r-ε)ν(H) for some ε>0. Further exact cases include maximum degree at most 2 and several strongly intersecting families. Unresolved remainder: Prove the conjectured coefficient r-1 for arbitrary r-partite, r-uniform hypergraphs when r≥4, or give a counterexample. Even the intersecting case ν(H)=1 remains open for every r≥6. The 2025 counterexample to the stronger Lovász conjecture does not settle Ryser’s conjecture. Corpus searches found only unrelated Bruck-Ryser references and no equivalent TheoremDB target.",
  "relevance": "Documents why Ryser’s conjecture for multipartite hypergraphs was treated as a distinct open target on 2026-08-01.",
  "relevance_source": "recorded",
  "body": "The audit ran on 2026-08-01 across the cited primary literature and problem collections, Crossref, arXiv, and stable publisher records, TheoremDB published, prospecting, packet, formal, and retired corpus. It checked the exact statement, its named or normalized variants, recent status evidence, and the controlled TheoremDB corpus.\n\nQueries included:\n- Ryser conjecture r-partite hypergraph open 2026\n- Ryser conjecture remains open for all r greater than or equal to 4\n- Ryser conjecture intersecting hypergraph r 6 open\n- Lovasz conjecture counterexample Ryser 2025\n\nThe exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: The conjecture is proved for all hypergraphs when r≤3 and for intersecting hypergraphs when r≤5. For r=4 and r=5, Haxell and Scott prove τ(H)≤(r-ε)ν(H) for some ε>0. Further exact cases include maximum degree at most 2 and several strongly intersecting families. Unresolved remainder: Prove the conjectured coefficient r-1 for arbitrary r-partite, r-uniform hypergraphs when r≥4, or give a counterexample. Even the intersecting case ν(H)=1 remains open for every r≥6. The 2025 counterexample to the stronger Lovász conjecture does not settle Ryser’s conjecture. Corpus searches found only unrelated Bruck-Ryser references and no equivalent TheoremDB target.",
  "status": "completed",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://doi.org/10.48550/arXiv.2505.05339",
      "locator": "A. Clow, P. Haxell, and B. Mohar, A Counterexample to a Conjecture of Lovász, arXiv:2505.05339 (2025). Abstract and Introduction"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.48550/arXiv.2505.05339",
    "locator": "A. Clow, P. Haxell, and B. Mohar, A Counterexample to a Conjecture of Lovász, arXiv:2505.05339 (2025). Abstract and Introduction"
  },
  "relations": [
    {
      "slug": "R1739",
      "title": "Current status and exact unresolved remainder",
      "object_type": "claim",
      "relation": "evidences",
      "direction": "outgoing"
    },
    {
      "slug": "ryser-hypergraph-cover-conjecture",
      "title": "ryser hypergraph cover conjecture",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}