{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1407",
  "content_hash": null,
  "slug": "borsuk-conjecture-in-four-dimensions-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 regular four-simplex gives b(4)≥5, while the current checked upper bound is b(4)≤8. Exact unresolved remainder: Determine whether b(4)=5; equivalently, prove five smaller-diameter parts always suffice or construct a unit-diameter set requiring at least six.",
  "relevance": "This is the dated publication status for the canonical target Borsuk’s conjecture in four dimensions.",
  "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 regular four-simplex gives b(4)≥5, while the current checked upper bound is b(4)≤8.\n\nThe exact unresolved remainder is: Determine whether b(4)=5; equivalently, prove five smaller-diameter parts always suffice or construct a unit-diameter set requiring at least six.\n\nA complete resolution must meet the following acceptance conditions:\n- For a proof, partition every bounded unit-diameter subset of R⁴ into five strictly smaller-diameter classes.\n- For a disproof, give an explicit finite or compact unit-diameter set whose diameter graph needs at least six colors.",
  "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.2605.19068",
      "locator": "A. Tolmachev and V. Voronov, “Reducing the upper bound for the Borsuk number in R⁴ to 8,” arXiv:2605.19068 (2026). abstract and main constructions"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.48550/arXiv.2605.19068",
    "locator": "A. Tolmachev and V. Voronov, “Reducing the upper bound for the Borsuk number in R⁴ to 8,” arXiv:2605.19068 (2026). abstract and main constructions"
  },
  "relations": [
    {
      "slug": "R1406",
      "title": "Strongest checked neighboring result",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R1404",
      "title": "Dated source and duplicate audit",
      "object_type": "attempt",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R1405",
      "title": "Work at the unresolved boundary",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
    },
    {
      "slug": "borsuk-conjecture-in-four-dimensions",
      "title": "borsuk conjecture in four dimensions",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}