{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1260",
  "content_hash": null,
  "slug": "erdos-problem-75-claim-status-20260731",
  "type": "claim",
  "title": "Current status and unresolved remainder",
  "summary": "OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 75 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 75: Does there exist a graph with chromatic number \\(\\aleph_1\\) and exactly \\(\\aleph_1\\) vertices such that for every real number \\(\\varepsilon > 0\\), there exists a natural number \\(N\\) with the following property: for every natural number \\(n \\geq N\\) and every subgraph \\(H\\) on exactly \\(n\\) vertices, there exists an independent set \\(I\\) contained in the vertex set of \\(H\\) whose cardinality exceeds \\(n^{1-\\varepsilon}\\)?",
  "relevance": "For erdos problem 75, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 75 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 75: Does.",
  "relevance_source": "recorded",
  "body": "OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 75 as open. The unresolved remainder is the full displayed statement.\n\nA complete resolution must satisfy this condition: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 75: Does there exist a graph with chromatic number \\(\\aleph_1\\) and exactly \\(\\aleph_1\\) vertices such that for every real number \\(\\varepsilon > 0\\), there exists a natural number \\(N\\) with the following property: for every natural number \\(n \\geq N\\) and every subgraph \\(H\\) on exactly \\(n\\) vertices, there exists an independent set \\(I\\) contained in the vertex set of \\(H\\) whose cardinality exceeds \\(n^{1-\\varepsilon}\\)?",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://www.erdosproblems.com/75",
      "locator": "See dataset.references[0] for the exact external source and locator."
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://www.erdosproblems.com/75",
    "locator": "See dataset.references[0] for the exact external source and locator."
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R1259",
      "title": "Resolve the stated acceptance condition",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
    },
    {
      "slug": "erdos-problem-75",
      "title": "erdos problem 75",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}