{
  "abstract": "We improve the bounds on the Hales-Jewett numbers to a tower of exponentiations. Earlier it was $WaW$ (that is, iterations of towers which are themselves iterated exponentiations). We improve the inductive step there (induction on the size of the alphabet, $|Λ|$) to 2-exponentiations, instead of towers. In the longer work in typing, (A) We present this inductive step as a partition theorem in its own right; (but in this preliminary version we make it just serve the bound on HJ numbers). (B) We shall deal with the density version of Hales-Jewett with similar bound. We are also dealing with the Graham-Rothschild Theorem and the Affine Ramsey Theorem and the polynomial case, and give background.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.14732",
  "author_total": 1,
  "authors": [
    {
      "name": "Saharon Shelah"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fc2etm74kwp3f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.14732",
  "fulltext_url": null,
  "id": "fc2etm74kwp3f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.14732",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7169503210",
  "pdf_url": "https://arxiv.org/pdf/2607.14732",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fc2etm74kwp3f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2607.14732",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.14732"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "HJ numbers revisited",
  "title_text": "HJ numbers revisited",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/fc2etm74kwp3f",
  "venue": "arXiv",
  "versions": []
}