{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1177",
  "content_hash": null,
  "slug": "erdos-problem-1-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 1: A finite set of real numbers $A$ is said to be a sum-distinct set for $N \\in \\mathbb{N}$ if $A \\subseteq (0, N]$ and the subset sums $\\sum_{a \\in S} a$ differ by at least $1$ for all distinct subsets $S \\subseteq A$. That is, for any two distinct subsets $S_1, S_2 \\subseteq A$, we have $1 \\leq |\\sum_{a \\in S_1} a - \\sum_{a \\in S_2} a|$. Determine whether there exists a positive real constant $C$ such that for every nonzero natural number $N$ and every finite set $A \\subseteq (0, N]$ that is sum-distinct for $N$, the inequality $C \\cdot 2^{|A|} < N$ holds.",
  "relevance": "For Erdős's sum-distinct set problem for real numbers, record erdos-problem-1-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 1: A finite set of real numbers $A$ is said to be a sum-distinct set for $N \\in \\mathbb{N}$ if $A \\subseteq (0, N]$ and the subset sums $\\sum_{a \\in S} a$ differ by at least $1$ for all distinct subsets $S \\subseteq A$.",
  "relevance_source": "recorded",
  "body": "Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 1: A finite set of real numbers $A$ is said to be a sum-distinct set for $N \\in \\mathbb{N}$ if $A \\subseteq (0, N]$ and the subset sums $\\sum_{a \\in S} a$ differ by at least $1$ for all distinct subsets $S \\subseteq A$. That is, for any two distinct subsets $S_1, S_2 \\subseteq A$, we have $1 \\leq |\\sum_{a \\in S_1} a - \\sum_{a \\in S_2} a|$. Determine whether there exists a positive real constant $C$ such that for every nonzero natural number $N$ and every finite set $A \\subseteq (0, N]$ that is sum-distinct for $N$, the inequality $C \\cdot 2^{|A|} < N$ 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/1",
      "locator": "Editorial research route recorded 2026-07-31"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://www.erdosproblems.com/1",
    "locator": "Editorial research route recorded 2026-07-31"
  },
  "models": [],
  "relations": [
    {
      "slug": "R1178",
      "title": "Current status and unresolved remainder",
      "object_type": "claim",
      "relation": "addresses",
      "direction": "outgoing"
    },
    {
      "slug": "erdos-problem-1",
      "title": "erdos problem 1",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}