{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1739",
  "content_hash": null,
  "slug": "ryser-hypergraph-cover-conjecture-claim-status-20260801",
  "type": "claim",
  "title": "Current status and exact unresolved remainder",
  "summary": "OPEN as checked on 2026-08-01. Strongest checked neighboring 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. Exact 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": "This is the dated publication status for the canonical target Ryser’s conjecture for multipartite hypergraphs.",
  "relevance_source": "recorded",
  "body": "The problem was checked as open on 2026-08-01.\n\nThe strongest neighboring result found in the cited sources is: 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\nThe exact unresolved remainder is: 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.\n\nA complete resolution must meet the following acceptance conditions:\n- Prove τ(H) ≤ (r-1)ν(H) for every finite r-partite, r-uniform hypergraph and every r≥2.\n- Or give an explicit r-partite, r-uniform hypergraph H with a certified matching number and transversal number satisfying τ(H)>(r-1)ν(H).",
  "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": "R1738",
      "title": "Strongest checked neighboring result",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R1736",
      "title": "Dated source and duplicate audit",
      "object_type": "attempt",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R1737",
      "title": "Work at the unresolved boundary",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
    },
    {
      "slug": "ryser-hypergraph-cover-conjecture",
      "title": "ryser hypergraph cover conjecture",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}