{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1816",
  "content_hash": null,
  "slug": "ms200-attempt-literature-and-exact-search-audit-reviewed-20260801",
  "type": "attempt",
  "title": "The literature gives asymptotics; the finite exact search remains incomplete",
  "summary": "Primary and modern sources study the asymptotic extremal function, while the focused audit found no exact table for n=200.",
  "relevance": "For Largest multiplicative Sidon subset of the first 200 integers, record ms200-attempt-literature-and-exact-search-audit (“The literature gives asymptotics; the finite exact search remains incomplete”) documents a concrete method, search boundary, or failed route. The record states: Primary and modern sources study the asymptotic extremal function, while the focused audit found no exact table for n=200.",
  "relevance_source": "recorded",
  "body": "Erdős introduced this extremal problem in 1938. His original paper gave a construction above \\(\\pi(n)\\) by a lower-order term and an upper bound \\(\\pi(n)+O(n^{3/4})\\). His 1969 paper sharpened the order, proving\n\\[\ns(n)=\\pi(n)+\\Theta\\!\\left(\\frac{n^{3/4}}{(\\log n)^{3/2}}\\right).\n\\]\nLiu and Pach restate this history in their 2018 paper and study the number of multiplicative Sidon subsets. Their introduction and bibliography identify the two primary Erdős papers. None of these sources tabulates \\(s(200)\\).\n\nA focused search for exact finite tables, the phrase `multiplicative Sidon`, and \\(s(200)\\) found asymptotic and generalized results. It found no published exact value at 200. A direct Boolean model of the 20,111 forbidden edges was also tried locally. It reproduced the 71 lower bound, while an attempted proof at the neighboring upper threshold did not finish within the allotted run. The retained LP computation gives a rigorous upper bound and records the remaining gap without claiming an exact solution.",
  "status": "completed",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://arxiv.org/abs/1808.06182",
      "locator": "Hong Liu and Péter Pál Pach, The number of multiplicative Sidon sets of integers, introduction, equation (1.1), and references [8] and [9]"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1808.06182",
    "locator": "Hong Liu and Péter Pál Pach, The number of multiplicative Sidon sets of integers, introduction, equation (1.1), and references [8] and [9]"
  },
  "models": [],
  "relations": [
    {
      "slug": "R539",
      "title": "The literature gives asymptotics; the finite exact search remains incomplete",
      "object_type": "attempt",
      "relation": "supersedes",
      "direction": "outgoing"
    },
    {
      "slug": "multiplicative-sidon-200",
      "title": "multiplicative sidon 200",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}