{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1235",
  "content_hash": null,
  "slug": "erdos-problem-39-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 39: A set $A$ of natural numbers is called a Sidon set if all pairwise sums of its elements are distinct, i.e., if $a + b = c + d$ with $a, b, c, d \\in A$, then $\\{a, b\\} = \\{c, d\\}$. Does there exist an infinite Sidon set $A \\subseteq \\mathbb{N}$ such that for every real number $\\varepsilon > 0$, the counting function satisfies\n\\[ |A \\cap \\{1, 2, \\ldots, N\\}| \\gg_\\varepsilon N^{1/2 - \\varepsilon} \\]\nas $N \\to \\infty$? Here, the notation $f(N) \\gg_\\varepsilon g(N)$ means that $g(N) = O(f(N))$ with the implied constant depending on $\\varepsilon$.",
  "relevance": "For Existence of dense infinite Sidon sets, record erdos-problem-39-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 39: A set $A$ of natural numbers is called a Sidon set if all pairwise sums of its elements are distinct, i.e., if $a + b = c + d$ with $a, b, c, d \\in A$, then $\\{a, b\\} = \\{c, d\\}$.",
  "relevance_source": "recorded",
  "body": "Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 39: A set $A$ of natural numbers is called a Sidon set if all pairwise sums of its elements are distinct, i.e., if $a + b = c + d$ with $a, b, c, d \\in A$, then $\\{a, b\\} = \\{c, d\\}$. Does there exist an infinite Sidon set $A \\subseteq \\mathbb{N}$ such that for every real number $\\varepsilon > 0$, the counting function satisfies\n\\[ |A \\cap \\{1, 2, \\ldots, N\\}| \\gg_\\varepsilon N^{1/2 - \\varepsilon} \\]\nas $N \\to \\infty$? Here, the notation $f(N) \\gg_\\varepsilon g(N)$ means that $g(N) = O(f(N))$ with the implied constant depending on $\\varepsilon$. 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/39",
      "locator": "Editorial research route recorded 2026-07-31"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://www.erdosproblems.com/39",
    "locator": "Editorial research route recorded 2026-07-31"
  },
  "models": [],
  "relations": [
    {
      "slug": "R1236",
      "title": "Current status and unresolved remainder",
      "object_type": "claim",
      "relation": "addresses",
      "direction": "outgoing"
    },
    {
      "slug": "erdos-problem-39",
      "title": "erdos problem 39",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}