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PreprintCC BY 4.0arXiv

HJ numbers revisited

  • Saharon Shelah
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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.

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Cite

BibTeXRIS
BibTeX
@misc{shelah2026numbers,
  title = {{HJ numbers revisited}},
  author = {Saharon Shelah},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.14732},
  eprint = {2607.14732},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.14732},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - HJ numbers revisited
AU  - Saharon Shelah
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2607.14732
UR  - https://arxiv.org/abs/2607.14732
AB  - 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.
ER  -