{
  "abstract": "The Schur number S(n) is the largest N such that {1, . . . , N } can be partitioned into n sum-free sets. Five values are known, 1, 4, 13, 44, 160, and the sixth is bracketed by 536 ≤ S(6) ≤ 1836. We define an integer sequence L(n) by a recurrence of Schur's type, L(n + 1) = 3 n + L(n) plus an alternating sum of floor corrections, which reproduces the five known values without adjustment, and we conjecture L(n) ≤ S(n) for all n. The corrections are finitely many: they vanish from the 25th on, and L(n) = (3 n -1)/2 + 22 542 -7(-1) n for all n ≥ 26. Given the published lower bounds and the recurrences of Rowley, of Ageron and co-authors and of Bengone and co-authors, the conjecture holds for all n ≥ 9 and is equivalent to the two statements S(6) ≥ 566 and S(7) ≥ 1852, both above the present records 536 and 1696. We list what a six-colouring of {1, . . . , w} would establish for each w between 537 and 620, the range of a search in progress. On the question of an upperbounding sequence of the same kind, we show that the only proven ones are factorial, S(n) ≤ ⌊n!(e -1/6)⌋ -1, and that the conjectural recursion of Eliahou and Revuelta is factorial too. Any exponential one must have a base at least 380 1/5 ≈ 3.28 and would settle an open problem on the growth of S(n). Two exponential candidates compatible with everything known are recorded with their refutation thresholds: U 1 (n) = 9 • 4 n-3 + 2 n-1 , which fits the three values 13, 44, 160 with two parameters and is refuted by a six-colouring of {1, . . . , 609}, and U 2 , defined by the recurrence U 2 (n) = 2 3 (4 n-1 +2)-U 2 (n-2)+U 2 (n-4) from the four values 1, 4, 13, 44, which predicts 160 and is refuted by a six-colouring of {1, . . . , 645}.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Patrick B. Reynaud"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bs3exy1sc5et0.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "bs3exy1sc5et0",
  "issns": [],
  "landing_url": "https://hal.science/hal-05836388v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.phys"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05836388/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bs3exy1sc5et0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:01Z",
      "native_id": "hal-05836388",
      "source": "hal",
      "url": "https://hal.science/hal-05836388v1"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "hal:0.phys",
      "subfield": "general"
    }
  ],
  "title": "A conjectured lower-bounding sequence for the Schur numbers, its reduction to S(6) and S(7), and the question of an upper-bounding sequence",
  "title_text": "A conjectured lower-bounding sequence for the Schur numbers, its reduction to S(6) and S(7), and the question of an upper-bounding sequence",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:01Z",
  "url": "https://research.draconitic.com/paper/bs3exy1sc5et0",
  "venue": "HAL",
  "versions": []
}