{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1233",
  "content_hash": null,
  "slug": "erdos-problem-36-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 36: For each positive integer $n$, consider all partitions of $\\{1, 2, \\dots, 2n\\}$ into two disjoint $n$-element subsets $A$ and $B$. For integers $a \\in A$ and $b \\in B$, define the overlap of $A$ and $B$ at difference $k \\in \\mathbb{Z}$ to be the number of pairs $(a,b)$ with $a - b = k$. Define the maximum overlap $M(A,B)$ to be the largest overlap over all integer differences $k$. Define $M(n)$ to be the minimum of $M(A,B)$ over all such partitions $(A,B)$. Define the minimum overlap quotient by\n\\[Q(n) = \\frac{M(n)}{n}.\\]\nDoes the limit $\\displaystyle\\lim_{n \\to \\infty} Q(n)$ exist, and if so, what is its exact value?",
  "relevance": "For The limit of the minimum overlap quotient, record erdos-problem-36-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 36: For each positive integer $n$, consider all partitions of $\\{1, 2, \\dots, 2n\\}$ into two disjoint $n$-element subsets $A$ and $B$.",
  "relevance_source": "recorded",
  "body": "Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 36: For each positive integer $n$, consider all partitions of $\\{1, 2, \\dots, 2n\\}$ into two disjoint $n$-element subsets $A$ and $B$. For integers $a \\in A$ and $b \\in B$, define the overlap of $A$ and $B$ at difference $k \\in \\mathbb{Z}$ to be the number of pairs $(a,b)$ with $a - b = k$. Define the maximum overlap $M(A,B)$ to be the largest overlap over all integer differences $k$. Define $M(n)$ to be the minimum of $M(A,B)$ over all such partitions $(A,B)$. Define the minimum overlap quotient by\n\\[Q(n) = \\frac{M(n)}{n}.\\]\nDoes the limit $\\displaystyle\\lim_{n \\to \\infty} Q(n)$ exist, and if so, what is its exact value? 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/36",
      "locator": "Editorial research route recorded 2026-07-31"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://www.erdosproblems.com/36",
    "locator": "Editorial research route recorded 2026-07-31"
  },
  "models": [],
  "relations": [
    {
      "slug": "R1234",
      "title": "Current status and unresolved remainder",
      "object_type": "claim",
      "relation": "addresses",
      "direction": "outgoing"
    },
    {
      "slug": "erdos-problem-36",
      "title": "erdos problem 36",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}