{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1738",
  "content_hash": null,
  "slug": "ryser-hypergraph-cover-conjecture-claim-literature-frontier",
  "type": "claim",
  "title": "Strongest checked neighboring result",
  "summary": "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.",
  "relevance": "Locates the present research frontier immediately below Ryser’s conjecture for multipartite hypergraphs.",
  "relevance_source": "recorded",
  "body": "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.\n\nThis leaves the following boundary unresolved: 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. The distinction is retained here so a restricted theorem, finite computation, or neighboring case is not presented as a solution of the full target.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "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": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "ryser-hypergraph-cover-conjecture",
      "title": "ryser hypergraph cover conjecture",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}