{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1255",
  "content_hash": null,
  "slug": "erdos-problem-70-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 70: Let $\\mathfrak{c}$ denote the cardinality of the continuum. For ordinals $\\alpha$, $\\beta$ and a cardinal $c$, the 3-uniform ordinal Ramsey property $\\alpha \\to (\\beta, c)^3_2$ is defined as follows: for every 2-coloring of the 3-element subsets of $\\alpha$ (identified with its ordinal type), either there exists a subset of order type $\\beta$ all of whose 3-element subsets are colored red, or there exists a subset of cardinality $c$ all of whose 3-element subsets are colored blue. The coloring is formally given as a symmetric predicate on ordered triples of distinct elements. Determine whether for every countable ordinal $\\beta$ and every integer $n \\geq 2$, the partition relation $\\mathfrak{c} \\to (\\beta, n)^3_2$ holds.",
  "relevance": "For The 3-uniform partition relation for the continuum, record erdos-problem-70-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 70: Let $\\mathfrak{c}$ denote the cardinality of the continuum.",
  "relevance_source": "recorded",
  "body": "Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 70: Let $\\mathfrak{c}$ denote the cardinality of the continuum. For ordinals $\\alpha$, $\\beta$ and a cardinal $c$, the 3-uniform ordinal Ramsey property $\\alpha \\to (\\beta, c)^3_2$ is defined as follows: for every 2-coloring of the 3-element subsets of $\\alpha$ (identified with its ordinal type), either there exists a subset of order type $\\beta$ all of whose 3-element subsets are colored red, or there exists a subset of cardinality $c$ all of whose 3-element subsets are colored blue. The coloring is formally given as a symmetric predicate on ordered triples of distinct elements. Determine whether for every countable ordinal $\\beta$ and every integer $n \\geq 2$, the partition relation $\\mathfrak{c} \\to (\\beta, n)^3_2$ holds. 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/70",
      "locator": "Editorial research route recorded 2026-07-31"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://www.erdosproblems.com/70",
    "locator": "Editorial research route recorded 2026-07-31"
  },
  "models": [],
  "relations": [
    {
      "slug": "R1256",
      "title": "Current status and unresolved remainder",
      "object_type": "claim",
      "relation": "addresses",
      "direction": "outgoing"
    },
    {
      "slug": "erdos-problem-70",
      "title": "erdos problem 70",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}