{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R717",
  "content_hash": null,
  "slug": "sdd50-claim-golomb-sidon-context",
  "type": "claim",
  "title": "The condition strengthens the Golomb ruler and Sidon property",
  "summary": "Distinct positive differences make every candidate a Golomb ruler; disjoint sum and difference value sets add a strict cross condition.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "Atkinson, Santoro, and Urrutia call an integer set with distinct nonzero differences a distinct difference set and connect this condition to distinct pair sums. Beck, Bogart, and Pham use the standard Golomb ruler description: a sequence of distinct integer marks whose pairwise differences are distinct. Over the integers this is the usual Sidon condition in difference form.\n\nEvery candidate set in the present problem is a Golomb ruler because its positive differences must be distinct. It also requires\n\\[\n\\{b-a:a<b,\\ a,b\\in S\\}\\cap\\{a+b:a<b,\\ a,b\\in S\\}=\\varnothing.\n\\]\nThis cross-disjointness is a strict addition. The Golomb ruler \\(\\{1,2,4\\}\\) fails it since \\(4-1=1+2=3\\). The DDS and Golomb ruler sources supply the parent concept without an exact value for this added finite problem. A focused source search performed for this entry found no published determination of \\(M(50)\\).",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "family",
    "statement": "finite increasing integer sets with distinct positive pairwise differences",
    "family": "Golomb rulers, distinct difference sets, and integer Sidon sets"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1109/TCOM.1986.1096587",
      "locator": "M. D. Atkinson, N. Santoro, and J. Urrutia, Integer Sets with Distinct Sums and Differences and Carrier Frequency Assignments for Nonlinear Repeaters, IEEE Transactions on Communications 34(6), 614-617 (1986), section II.A, DOI 10.1109/TCOM.1986.1096587; Matthias Beck, Tristram Bogart, and Tu Pham, Enumeration of Golomb Rulers and Acyclic Orientations of Mixed Graphs, Electronic Journal of Combinatorics 19(3) (2012), P42, arXiv:1110.6154"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1109/TCOM.1986.1096587",
    "locator": "M. D. Atkinson, N. Santoro, and J. Urrutia, Integer Sets with Distinct Sums and Differences and Carrier Frequency Assignments for Nonlinear Repeaters, IEEE Transactions on Communications 34(6), 614-617 (1986), section II.A, DOI 10.1109/TCOM.1986.1096587; Matthias Beck, Tristram Bogart, and Tu Pham, Enumeration of Golomb Rulers and Acyclic Orientations of Mixed Graphs, Electronic Journal of Combinatorics 19(3) (2012), P42, arXiv:1110.6154"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R716",
      "title": "The exact maximum is M(50)=8",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "sum-difference-distinct-subset-50",
      "title": "sum difference distinct subset 50",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}