{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1223",
  "content_hash": null,
  "slug": "erdos-problem-28-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 28: Let $A$ be a set of natural numbers. For each $n \\in \\mathbb{N}$, define the representation function $r_A(n)$ to be the number of ordered pairs $(a, b) \\in A \\times A$ such that $a + b = n$. Suppose that the complement of the sumset $A + A = \\{a + b : a, b \\in A\\}$ is finite, meaning that all but finitely many natural numbers can be expressed as a sum of two elements of $A$. Prove that $\\limsup_{n \\to \\infty} r_A(n) = \\infty$.",
  "relevance": "For Unbounded representation function for sets with cofinite sumset, record erdos-problem-28-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 28: Let $A$ be a set of natural numbers.",
  "relevance_source": "recorded",
  "body": "Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 28: Let $A$ be a set of natural numbers. For each $n \\in \\mathbb{N}$, define the representation function $r_A(n)$ to be the number of ordered pairs $(a, b) \\in A \\times A$ such that $a + b = n$. Suppose that the complement of the sumset $A + A = \\{a + b : a, b \\in A\\}$ is finite, meaning that all but finitely many natural numbers can be expressed as a sum of two elements of $A$. Prove that $\\limsup_{n \\to \\infty} r_A(n) = \\infty$. 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/28",
      "locator": "Editorial research route recorded 2026-07-31"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://www.erdosproblems.com/28",
    "locator": "Editorial research route recorded 2026-07-31"
  },
  "models": [],
  "relations": [
    {
      "slug": "R1224",
      "title": "Current status and unresolved remainder",
      "object_type": "claim",
      "relation": "addresses",
      "direction": "outgoing"
    },
    {
      "slug": "erdos-problem-28",
      "title": "erdos problem 28",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}