HJ numbers revisited

Saharon Shelah

HJ numbers revisited

Combinatorics 2026-07-16 v1

Authors: Saharon Shelah

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, $|\Lambda|$ ) 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.

Cite

@article{arxiv.2607.14732,
  title  = {HJ numbers revisited},
  author = {Saharon Shelah},
  journal= {arXiv preprint arXiv:2607.14732},
  year   = {2026}
}