{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1240",
  "content_hash": null,
  "slug": "erdos-problem-41-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 41 as open. The unresolved remainder is the full displayed statement. 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 erdos problem 41, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 41 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 41: A.",
  "relevance_source": "recorded",
  "body": "OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 41 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 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 $0$?",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://www.erdosproblems.com/41",
      "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/41",
    "locator": "See dataset.references[0] for the exact external source and locator."
  },
  "models": [],
  "relations": [
    {
      "slug": "R1239",
      "title": "Resolve the stated acceptance condition",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
    },
    {
      "slug": "erdos-problem-41",
      "title": "erdos problem 41",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}