{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1239",
  "content_hash": null,
  "slug": "erdos-problem-41-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 41: A set $A \\subseteq \\mathbb{N}$ is said to satisfy the triple distinct sums condition if for any two finite subsets $I, J \\subseteq A$ with $|I| = |J| = 3$, the equality $\\sum_{i \\in I} i = \\sum_{j \\in J} j$ implies $I = J$. In other words, all sums of three elements from $A$ are distinct aside from trivial coincidences arising from reordering the same three elements. Let $A \\subseteq \\mathbb{N}$ be an infinite set satisfying the triple distinct sums condition. For each positive integer $N$, let $|A \\cap \\{1, \\ldots, N\\}|$ denote the number of elements of $A$ that are at most $N$. Does the lower limit\n\\[\n\\liminf_{N \\to \\infty} \\frac{|A \\cap \\{1, \\ldots, N\\}|}{N^{1/3}}\n\\]\nequal $0$?",
  "relevance": "For Erdős Problem 41 on Distinct Triple Sums, record erdos-problem-41-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 41: A set $A \\subseteq \\mathbb{N}$ is said to satisfy the triple distinct sums condition if for any two finite subsets $I, J \\subseteq A$ with $|I| = |J| = 3$, the equality $\\sum_{i \\in I} i = \\sum_{j \\in J} j$ implies $I = J$.",
  "relevance_source": "recorded",
  "body": "Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 41: A set $A \\subseteq \\mathbb{N}$ is said to satisfy the triple distinct sums condition if for any two finite subsets $I, J \\subseteq A$ with $|I| = |J| = 3$, the equality $\\sum_{i \\in I} i = \\sum_{j \\in J} j$ implies $I = J$. In other words, all sums of three elements from $A$ are distinct aside from trivial coincidences arising from reordering the same three elements. Let $A \\subseteq \\mathbb{N}$ be an infinite set satisfying the triple distinct sums condition. For each positive integer $N$, let $|A \\cap \\{1, \\ldots, N\\}|$ denote the number of elements of $A$ that are at most $N$. Does the lower limit\n\\[\n\\liminf_{N \\to \\infty} \\frac{|A \\cap \\{1, \\ldots, N\\}|}{N^{1/3}}\n\\]\nequal $0$? 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/41",
      "locator": "Editorial research route recorded 2026-07-31"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://www.erdosproblems.com/41",
    "locator": "Editorial research route recorded 2026-07-31"
  },
  "models": [],
  "relations": [
    {
      "slug": "R1240",
      "title": "Current status and unresolved remainder",
      "object_type": "claim",
      "relation": "addresses",
      "direction": "outgoing"
    },
    {
      "slug": "erdos-problem-41",
      "title": "erdos problem 41",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}