{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1259",
  "content_hash": null,
  "slug": "erdos-problem-75-attempt-resolution-route",
  "type": "attempt",
  "title": "Resolve the stated acceptance condition",
  "summary": "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 Erdős's problem on uncountable graphs with large independent sets in all large finite subgraphs, record erdos-problem-75-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: 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_source": "recorded",
  "body": "Target the displayed statement directly. 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}\\)? Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress.",
  "status": "open_strategy",
  "evidence_grade": "self_reported",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://www.erdosproblems.com/75",
      "locator": "Editorial research route recorded 2026-07-31"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://www.erdosproblems.com/75",
    "locator": "Editorial research route recorded 2026-07-31"
  },
  "models": [],
  "relations": [
    {
      "slug": "R1260",
      "title": "Current status and unresolved remainder",
      "object_type": "claim",
      "relation": "addresses",
      "direction": "outgoing"
    },
    {
      "slug": "erdos-problem-75",
      "title": "erdos problem 75",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}