{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1269",
  "content_hash": null,
  "slug": "erdos-problem-91-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 91: For a finite set $A\\subset\\mathbb{R}^2$, let $d(A)$ denote the number of distinct Euclidean distances between pairs of distinct points of $A$. For each positive integer $n$, define $m(n)$ to be the minimum value of $d(A)$ over all sets $A\\subset\\mathbb{R}^2$ with $|A|=n$. A set $A$ with $|A|=n$ and $d(A)=m(n)$ is called optimal (for $n$). Two finite sets $A,B\\subset\\mathbb{R}^2$ are called similar if there exists a dilation of the plane mapping $A$ onto $B$ (where a dilation is a composition of a scaling and an isometry). For each $n$, we say that $n$ has a unique minimizer if all optimal sets for $n$ are similar to one another. Determine whether, for all sufficiently large integers $n$, the number $n$ does not have a unique minimizer.",
  "relevance": "For Non-uniqueness of point sets minimizing distinct distances, record erdos-problem-91-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 91: For a finite set $A\\subset\\mathbb{R}^2$, let $d(A)$ denote the number of distinct Euclidean distances between pairs of distinct points of $A$.",
  "relevance_source": "recorded",
  "body": "Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 91: For a finite set $A\\subset\\mathbb{R}^2$, let $d(A)$ denote the number of distinct Euclidean distances between pairs of distinct points of $A$. For each positive integer $n$, define $m(n)$ to be the minimum value of $d(A)$ over all sets $A\\subset\\mathbb{R}^2$ with $|A|=n$. A set $A$ with $|A|=n$ and $d(A)=m(n)$ is called optimal (for $n$). Two finite sets $A,B\\subset\\mathbb{R}^2$ are called similar if there exists a dilation of the plane mapping $A$ onto $B$ (where a dilation is a composition of a scaling and an isometry). For each $n$, we say that $n$ has a unique minimizer if all optimal sets for $n$ are similar to one another. Determine whether, for all sufficiently large integers $n$, the number $n$ does not have a unique minimizer. 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/91",
      "locator": "Editorial research route recorded 2026-07-31"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://www.erdosproblems.com/91",
    "locator": "Editorial research route recorded 2026-07-31"
  },
  "models": [],
  "relations": [
    {
      "slug": "R1270",
      "title": "Current status and unresolved remainder",
      "object_type": "claim",
      "relation": "addresses",
      "direction": "outgoing"
    },
    {
      "slug": "erdos-problem-91",
      "title": "erdos problem 91",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}